Extension of the Wilson model to electrolyte solutions

Fluid Phase Equilibria 173 (2000) 161–175

Extension of the Wilson model to electrolyte solutions Ensheng Zhao, Ming Yu, Robert E. Sauvé, Mohammad K. Khoshkbarchi∗ AEA Technology Engineering Software-Hyprotech, Suite 800, 707-8th Avenue S.W. Calgary, Alta., Canada T2P 1H5 Received 25 January 2000; accepted 30 May 2000

Abstract The Wilson model for non-electrolytes has been extended to model the activity coefficients of electrolytes in aqueous solutions. The excess Gibbs energy of an aqueous electrolyte solution is expressed as a sum of contributions of a long-range and a short-range excess Gibbs energy term. The contribution of the long-range excess Gibbs energy is represented by the Pitzer–Debye–Hückel model. A new expression based on the local composition concept has been developed to account for the contribution of the short-range excess Gibbs energy. The existence of three types of local cells with a central cation, anion or solvent molecule is assumed. Due to the strong like-ion repulsion it is assumed that only counterions can be found in the immediate neighborhood of a central ion. The main difference between this model and the extensions of the NRTL model to electrolytes available in the literature is the assumption that the short-range energy parameter between species in a local cell has an enthalpic rather than Gibbs energy nature. For systems containing only molecular components the new model simplifies to the three-parameter version of the Wilson model. The model has been applied to several single electrolyte systems and it has been shown that it can represent the mean ionic activity coefficients with good accuracy. © 2000 Elsevier Science B.V. All rights reserved. Keywords: Model; Wilson; Activity coefficient; Electrolyte solutions

1. Introduction Electrolyte solutions participate in many chemical and environmental processes. Accurate models for the thermodynamic properties of electrolyte solutions are essential for the design and control of these processes and are extensively used by process simulators. However, development of rigorous thermodynamic models for electrolytes, due to the complex nature of their behavior in solutions, is a cumbersome task. Furthermore, due to the complexity of the rigorous treatment of electrolyte solutions, their application for process design and simulation purposes is not feasible. Therefore, many attempts have been made to develop models with simpler expressions for the thermodynamic properties of electrolyte solutions. ∗ Corresponding author. Tel.: +1-403-520-6000; fax: +1-403-520-6060 E-mail address: [email protected] (M.K. Khoshkbarchi).

0378-3812/00/$20.00 © 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 3 8 1 2 ( 0 0 ) 0 0 3 9 3 - 9

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The empirical and semi-empirical extensions of the Debye–Hückel model have provided the most attractive frameworks for the development of many of the models for electrolyte solutions. These models usually comprise of a long-range electrostatic interaction term represented by the Debye–Hückel model and a residual term, which accounts for all the short-range effects. The most widely used forms of these models are those based on the Pitzer formalism [1,2] and local composition concept [3–7]. In the Pitzer formalism, the short-range interactions are represented by a virial expansion type equation, while in the second class of models an expression based on the local composition concept is employed to account for the short-range interactions. Among the models based on the local composition concept, extensions of the NRTL model [8] to electrolyte solutions have received a great deal of attention and proven to be useful for simulation purposes [3,4,9]. In these models, similar to the NRTL model, it is assumed that the short-range energy parameter in a local cell is proportional to the local Gibbs energy. The Wilson model [10] has been widely used to represent the non-idealities in mixtures of nonelectrolytes. The Wilson model was originally developed by incorporating the energy of interactions among molecules to the Flory–Huggins theory [11–13] framework. In the original derivation of the Wilson model, the Gibbs energy of mixing, GM , is related to the local volume fraction ξ , as GM X xi ln ξi = RT i

(1)

The extension of the Wilson equation to electrolyte solutions using the method proposed by Wilson [10], either requires several adjustable parameters, or leads to a singularity in the resulting expression. Renon and Prausnitz [14] presented an alternative approach to derive the Wilson model and showed that the Wilson model, similar to the NRTL model, could be derived from the local composition concept assuming that the energy parameter is equivalent to the local enthalpy of the cells. It was also shown that this approach can introduce a third parameter to the original Wilson model. The third parameter is similar to the coordination number parameter proposed by the Guggenheim’s quasichemical theory (1945). As discussed by Wilson [10] the three-parameter Wilson model, unlike its two-parameter version, is capable of predicting phase split in liquids. In this study, we have extended the Wilson model to electrolyte solutions. Similar to the extensions of the NRTL model to electrolyte solutions [4,9,15,16], the long-range interactions are represented by the Pitzer–Debye–Hückel model [1,2]. However, in contrast to the previous models, the short-range interactions in our model are expressed by a local composition model, which for molecular components simplifies to the Wilson model.

2. Thermodynamic framework The excess Gibbs energy of an electrolyte solution is assumed to be a sum of the contributions of a long-range electrostatic term, GE,LR , and a short-range interaction term, GE,SR , as GE,LR GE,SR GE = + RT RT RT

(2)

The long-range interaction term accounts for the electrostatic interactions between ions and the short-range interaction term considers the ion–solvent interactions as well as the non-electrostatic ion–ion interactions.

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163

To represent the long-range interaction term we use the extended version of the Debye–Hückel model, GE,PDH , proposed by Pitzer [19]: √ X  1000 1/2 4Aφ Ix ln(1 + ρ Ix ) GE,PDH = − xi (3) RT MS ρ i where, Aφ represents the Debye–Hückel constant for the osmotic coefficient, MS is the molecular weight of the solvent, ρ is the closest approach parameter and Ix is the ionic strength in mole fraction scale defined as 1X (4) Ix = xi Zi2 2 i with Zi being the charge number of ionic species. In this study, the short-range interactions are represented by a new expression derived from the local composition concept. Based on the local composition concept, the distribution of a molecule j about a central molecule i can be approximated by the following relation [17]: nji = nj e−Wji /RT

(5)

where, Wj i is the potential of mean force and n denotes the number of molecules. Eq. (5) for a multicomponent system, assuming that the potential of mean force is equivalent to the energy parameter between the species in a local cell, can be written as xji xj e−εji /RT = xk e−εki /RT xki

(6)

where xj i is the local mole fraction of molecule j around molecule i, and εj i is an energy parameter proportional to the interaction energy between molecules j and i. Renon and Prausnitz [8] assumed that the energy parameter ε in Eq. (6) is equivalent to the local Gibbs energy and developed the NRTL model for non-electrolytes. Later they showed that although the original Wilson model was derived from the Flory–Huggins theory [11–13], alternatively it could be derived from Eq. (6) provided that it is assumed that the energy parameter ε is equivalent to the local enthalpy [14]. Using this approach, a parameter similar to the coordination number in Guggenheim’s quasichemical theory [18], can also be introduced to the Wilson model. With this assumption Eq. (6) can be written as xji xj e−hji /CRT = xki xk e−hki /CRT

(7)

where C is the coordination number and hj i is the enthalpy of interaction between molecules j and i. 3. A new expression for the short-range interactions In an electrolyte solution the short-range forces are originated from the solvent-solvent, solvent–ion and non-electrostatic ion–ion interactions. To consider these interactions and similar to the approach of Chen et al. [4,9] we assume the existence of three types of local cells, namely, cells with a central cation, cells with a central anion and cells with a central solvent molecule. According to the specific interaction

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theory it is assumed that the like-ion repulsion is such strong that cells with a central cation do not contain any cations and cells with a central anion do not contain any anions. Since based on these assumptions the cells with a central ion contain only counterions and solvent molecules, they are not necessarily electrically neutral. However, cells with a central solvent molecule contain cations, anions and solvent molecules in the immediate neighborhood of the central solvent molecule. Therefore, cells with a central solvent molecule are electrically neutral. In a local cell the effective local mole fractions of species j and i in the neighborhood of species i, respectively denoted by Xj i and Xii , can be related to the effective global mole fractions of species j and i, respectively denoted by Xj and Xi , through   Xji Xj = (8) Hji Xii Xi where the effective mole fraction for an electrolyte system containing ion j carrying Zj charges is defined as ( for ions Kj = Zj Xj = xj Kj with (9) for molecules Kj = 1 In Eq. (9) xj is the apparent mole fraction of species j which fulfils the following condition: X xj = 1

(10)

j

where the sum runs over all molecular and ionic species. In a major departure from the extensions of the NRTL model to electrolytes, the parameter Hj i in Eq. (8) is related to the local cell enthalpies by Hji = e−(hji −hii )/CRT

(11)

where C is a parameter, which represents the effective coordination number in the system. The effective coordination number globally accounts for the effect of the size and charge differences of components on the coordination number of a multicomponent system. Using this approach the effective local mole fractions of species j and k in the neighborhood of ionic species i, respectively denoted by, Xj i and Xki , are related by     Xji Xj Hji Xj = = (12) Hji,ki Xki Xk Hki Xk where Hji,ki = e−(hji −hki )/CRT

(13)

The effective local mole fractions, for a system containing completely dissociated electrolytes in a single solvent, are related through the following relations: X X Xck m + Xak m + Xmm = 1 (central solvent cells) (14) k

k

Xmck

X + Xaj ck = 1

E. Zhao et al. / Fluid Phase Equilibria 173 (2000) 161–175

165

(central cation of electrolyte k cells)

(15)

(central anion of electrolyte k cells)

(16)

j

Xmak +

X Xcj ak = 1 j

where the subscripts ‘m’, ‘a’ and ‘c’ denote the solvent, anion and cation, respectively. It should be noted that based on the like-ion repulsion assumption Xcc = Xaa = 0. Combining Eqs. (8) to (16) the effective local mole fractions can be expressed in terms of the effective mole fractions as • Effective local mole fractions of ionic species i in a local cell with a central solvent molecule m: Xim =

Xm +

P

Xi Him P X k ck Hck m + k Xak Hak m

(17)

• Effective local mole fractions of anion of electrolyte k in a local cell with a central cation: Xak c =

Xm Hmc,ak c +

Xak l Xal (Hmc,ak c /Hmc,al c )

P

(18)

• Effective local mole fractions of cation of electrolyte k in a local cells with a central anion: Xck a =

Xm Hma,ck a +

P

Xck l Xcl (Hma,ck a /Hma,cl a )

(19)

The enthalpy of cells with a central cation of electrolyte k, anion of electrolyte k and solvent m, respectively denoted by hck , hak , hm , can be related to the effective local mole fractions as ! X ck h = Zck Xmck hmck + Xal ck hal ck (20) l

h = Zak Xmak hmak ak

X + Xcl ak hcl ak

! (21)

l

hm = Xmm hmm +

X X Xck m hck m + Xak m hak m k

(22)

k

The excess enthalpy of each of the local cells can then be defined with respect to a reference state enthalpy as hE,m = hm − hm ref

(23)

hE,ck = hck − hcrefk

(24)

hE,ak = hak − harefk

(25)

where hcrefk , harefk , hm ref are the reference enthalpies of the cells with a central cation of electrolyte k, a central anion of electrolyte k and a central solvent m, respectively. These reference enthalpies for the cations and the anions are considered to be the enthalpy of the imaginary completely dissociated fused electrolyte

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k and for the solvent as its enthalpy in the pure state. Based on these definitions the molar reference enthalpies can be written as hcrefk = Zck hak ck

(26)

harefk = Zak hck ak

(27)

hm ref = hmm

(28)

The excess enthalpy of the system, hE , can be calculated by summing the excess enthalpies of all the cells as X X xck (hck − hcrefk ) + xak (hak − harefk ) (29) hE = xm (hm − hm ref ) + k

k

Combining Eqs. (17) to (26) the following expression for the excess enthalpy of the system is obtained: X Xc Hc m (hc m − hmm ) + Xa Ha m (ha m − hmm ) k k k k k P Pk hE = Xm Xm + l Xcl Hcl m + l Xal Hal m k # " X X Xal (hal ck − hmck ) P + Xck (hmck − hak ck ) + Xm Hmck ,al ck + j Xaj (Hmck ,al ck /Hmck ,aj ck ) k l # " X X Xcl (hcl ak − hmak ) P (30) + Xak (hmak − hck ak ) + X H + X (H /H ) m ma ,c a c ma ,c a ma ,c a k l k j k l k k j k j k l To simplify the mathematical expression of Eq. (30) we introduce the parameter E as Eji = hji − hii

(31)

Eji,ki = hji − hki

(32)

In order to reduce the number of adjustable parameters in the final form of the model we assume Eck m = Eak m = Eek m

(33)

Emck ,ak ck = Emak ,ck ak = Em,ek

(34)

Eal ck = Ecl ak = Eel ek

(35)

where the subscript e refers to an electrolyte and it should be noted that Eel ek 6= Eek el for electrolyte k 6= l. Using the above assumptions and notations, Eq. (30) can be written as: P j (Xcj + Xaj )Eej m exp(−Eej m /CRT) E P h = Xm Xm + j (Xcj + Xaj ) exp(−Eej m /CRT) # " P X j Xaj (Eej ek − Emek ) exp(−(Eej ek − Emek )/CRT) P + Xck Em,ek + X + m j Xaj exp(−(Eej ek − Emek )/CRT) k # " P X j Xcj (Eej ek − Emek ) exp(−(Eej ek − Emek )/CRT) P (36) + Xak Em,ek + Xm + j Xcj exp(−(Eej ek − Emek )/CRT) k

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The excess Gibbs energy can then be obtained by combining Eq. (36) and the following exact thermodynamic relation:   Z 1 1 1/T E GE h d = (37) RT R 0 T where T is the absolute temperature and R is the universal gas constant. The excess Gibbs energy thus obtained is then normalized in the Henry’s convention using the following relation: GE∗ GE X (xck ln γc∞ + xak ln γa∞ ) = − k k RT RT k

(38)

, γa∞ are, respectively the infinite dilution activity where the sum runs over all ionic species and γc∞ k k coefficients of the cation and anion of electrolyte k, defined as   ∂(nGE /RT) ∞ lnγck = Plim = Zck C[−exp(−Eek m /CRT) + 1 + Emek /CRT] (39) xc →0 ∂nck T ,nj 6=ck   ∂(nGE /RT) ∞ = Zak C[−exp(−Eek m /CRT) + 1 + Emek /CRT] (40) lnγak = Plim xa →0 ∂nak T ,nj 6=ak where n is the number of molecules and GE∗ denotes the excess Gibbs energy normalized in Henry’s convention. The expression for the unsymmetric excess Gibbs energy is obtained as P   Xm + k (Xck + Xak )Hek m GE∗ P = −CXm ln RT Xm + k (Xck + Xak ) " ! # P X Xm Hek m + j Xaj Hej ek P − Hek m + 1 − ln Hek m −C Xck ln X + X m a j j k " ! # P X Xm Hek m + j Xcj Hej ek P − Hek m + 1 − ln Hek m (41) −C Xak ln X m + j X cj k The activity coefficients of the ionic and molecular species can be obtained by combining Eq. (41) and the following exact thermodynamic relation:   ∂(nGE /RT) (42) ln γi = ∂ni T ,nj 6=ni The expressions for the activity coefficients of the ionic and molecular species are given in the Appendix A. The mean ionic activity coefficient of an electrolyte with cation c and anion a, γ ±ca , can be calculated using the following relation: ln γ±ca =

νc ln γc + νa ln γa νc + νa

(43)

where, ν c and ν a are the stoichiometric numbers of the cation and the anion of the electrolyte, respectively.

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4. Results and discussion The model developed in this study was applied to correlate the mean ionic activity coefficients of several single electrolytes in aqueous solutions. The parameters of the model were evaluated using the following procedure. The Pitzer–Debye–Hückel model requires the evaluation of two parameters, the Debye–Hückel constant for osmotic coefficient, Aφ , which at 298.15 K is equal to 0.390947, and closest approach parameter ρ. Although in principle the value of the parameter ρ depends on the nature of the electrolyte, as suggested by Pitzer [19] it can be set to a constant value of 14.9 for all electrolytes. In order to reduce the number of the adjustable parameters we adopt this value for the parameter ρ of all the electrolytes. The short-range interaction term requires the evaluation of the energy parameters E and the coordination parameter C. The value of the C parameter in this model is set to 10 and the value of the energy parameters for single electrolyte systems is obtained by fitting the model to the experimental mean ionic activity coefficient data. In all cases the following objective function was minimized: OBJ =

NP X

exp

(ln γ±cal − ln γ± )2

(44)

i

where NP is the number of the data points and superscripts ‘cal’ and ‘exp’ refer to the calculated and the experimental values, respectively. As the experimental data available in the literature are normalized in molality scale and the activity coefficients calculated from our model are normalized in mole fraction scale, the following conversion was used: ! X x m ln γ± = ln γ± + ln 1 + 0.001MS (νck + νak )mk (45) k

Table 1 presents the energy parameters and the standard deviations obtained from the fitting of the new model to the experimental data of mean ionic activity coefficients of 129 different electrolytes in aqueous solutions. The standard deviation, σ , is defined as: "P #1/2 exp (ln γ± − ln γ±calc )2 σ = (46) NP The standard deviations of the fit obtained from the application of the model developed in this study and those obtained from the NRTL-model of electrolytes developed by Chen et al. [4] and Haghtalab and Vera [16] are also compared in Table 1. As presented in this table the model developed here correlates the experimental data with comparable and in most cases with better accuracy than the other two models. Table 2 presents the comparison of the average standard deviations obtained frsom these. The results indicate that, with the same number of parameters (i.e. two per water–electrolyte system), our new model can correlate the experimental data with better accuracy than the other two models. As mentioned before, for all electrolytes the value of the parameter C was set to 10. It is interesting to note that this parameter can also be treated as an adjustable parameter in which case the model will provide a better fit for the experimental data. Calculations were also performed to correlate the mean ionic activity coefficient data of the systems shown in Table 1 with the three-parameter version of our model. The comparison of

E. Zhao et al. / Fluid Phase Equilibria 173 (2000) 161–175

169

Table 1 Values of the energy parameters and standard deviations of the fit of Wilson-electrolyte model, NRTL–electrolyte model [4] and NRTL–NRF model [16] to mean ionic activity coefficient data at 298.15 K [20] with C = 10 Electrolyte

Maximum molality

Uni-univalent electrolytes AgNO3 6.0 CsAc 3.5 CsBr 5.0 CsCl 6.0 CsI 3.0 CsNO3 1.4 HBr 3.0 HCl 6.0 HClO4 6.0 HI 3.0 HNO3 3.0 Kac 3.5 KBr 5.5 KCl 4.5 KCNS 5.0 KF 4.0 KH Adipate 1.0 KH Malonate 5.0 KH Succinate 4.5 KH2 PO4 1.8 KI 4.5 KNO3 3.5 KOH 6.0 LiAc 4.0 LiBr 6.0 LiCl 6.0 LiClO4 4.0 LiI 3.0 LiNO3 6.0 LiOH 4.0 LiTol 4.5 NaBr 4.0 NaBrO3 2.5 Na Butyrate 3.5 Na Caproate 1.8 NaCl 6.0 NaClO3 3.5 NaClO4 6.0 NaCNS 4.0 NaF 1.0 Na Formate 3.5 NaH Malonate 5.0 NaH Succinate 5.0 NaH2 PO4 6.0 NaI 3.5

Eme (J/mol)

Eem (J/mol)

ρ (Wilson)

ρ (NRTL)

ρ (NRTL–NRF)

52382.50 6433.98 56279.08 56913.58 54345.16 56503.31 −103946.44 −97085.53 −108869.98 −12923.82 924.30 5923.81 −8890.70 −15757.89 −768.81 −61314.07 −6304.68 −24446.80 −39864.84 56462.17 4475.41 51947.41 −74536.51 −3866.55 −109858.87 −58396.95 −20449.08 −12699.10 −8835.22 58111.98 −4249.58 −24037.53 50903.57 50920.96 71674.65 −39738.43 1966.39 −4717.12 8895.55 −1953.66 4475.41 51947.41 −74536.51 −3866.55 −109858.87

−24740.93 −10803.12 −27767.45 −28100.81 −26798.53 −26799.37 −3295.87 −1197.66 −3009.92 −9776.94 −7400.75 −10283.06 −8890.70 3567.57 −1978.85 8954.54 −490.89 18139.79 20446.24 −26769.08 −6111.70 −24686.05 1445.65 −4854.05 −1981.14 −914.98 −7123.33 −8718.66 −5222.23 −28526.40 −1022.25 396.81 −25104.65 −27479.50 −31122.12 5616.28 −2733.95 −1875.64 −9511.51 −54.48 −6111.70 −24686.05 1445.65 −4854.05 −1981.14

0.0050 0.0088 0.0030 0.0078 0.0044 0.0028 0.0103 0.0176 0.0665 0.0143 0.0074 0.0070 0.0020 0.0010 0.0017 0.0023 0.0029 0.0076 0.0090 0.0027 0.0045 0.0049 0.0210 0.0038 0.0520 0.0314 0.0167 0.0214 0.0056 0.0198 0.0127 0.0054 0.0021 0.0052 0.0242 0.0082 0.0045 0.0072 0.0101 0.0006 0.0045 0.0032 0.0038 0.0134 0.0077

0.0100 0.0100 0.0060 0.0060 0.0070 0.0030 0.0150 0.0350 0.0630 0.0180 0.0080 0.0080 0.0040 0.0030 0.0020 0.0050 0.0020 0.0050 0.0030 0.0040 0.0050 0.0080 0.0230 0.0050 0.0500 0.0400 0.0230 0.0240 0.0130 0.0260 0.0130 0.0090 0.0020 0.0070 0.0240 0.0180 0.0050 0.0100 0.0100 0.0000 0.0040 0.0020 0.0020 0.0030 0.0100

0.014a 0.008 0.005 0.013a 0.005 0.000 0.040a 0.024a 0.026 0.062a 0.018a 0.008 0.004 0.021a 0.003 0.006a 0.003 0.004 0.003 0.002 0.005 0.004 0.039a 0.005 0.045 0.052a 0.017a 0.020 0.016a 0.021a 0.014 0.061a 0.002a 0.026 0.015a 0.011a 0.005b 0.009 0.039a 0.002 0.008 0.001 0.002 0.003a 0.028a

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Table 1 (Continued) Eem (J/mol)

ρ (Wilson)

ρ (NRTL)

ρ (NRTL–NRF)

−58396.95 −20449.08 −12699.10 −8835.22 58111.98 −4249.58 −24037.53 50903.57 50920.96 71674.65 −39738.43 1966.39

−914.98 −7123.33 −8718.66 −5222.23 −28526.40 −1022.25 396.81 −25104.65 −27479.50 −31122.12 5616.28 −2733.95

0.0021 0.0176 0.0448 0.0061 0.0019 0.0075 0.0073 0.0048 0.0053 0.0075 0.0070 0.0039

0.0020 0.0290 0.0420 0.0060 0.0010 0.0120 0.0090 0.0020 0.0020 0.0030 0.0120 0.0120

0.072a 0.057a 0.058 0.008 0.002a 0.010a 0.008 0.001 0.003a 0.002 0.007 0.010

Uni-divalent electrolytes Cs2 SO4 1.8 K2 CrO4 3.5 K2 SO4 0.7 Li2 SO4 3.0 Na2 CrO4 4.0 Na2 Fumarate 2.0 Na2 Maleate 3.0 Na2 SO4 4.0 Na2 S2 O3 3.5 4.0 (NH4 )2 SO4 Rb2SO4 1.8

−575.152 −19563.6 −31039.3 −12816.6 −63684.3 −26635.9 59685.66 −48013.9 −63221.7 −38094.9 −7741.79

−978.942 7932.383 48212.52 863.8951 17954.79 2221.519 −29304.6 64821.3 25743.51 55139.34 3993.361

0.0077 0.0140 0.0147 0.0157 0.0358 0.0047 0.0148 0.0263 0.0168 0.0561 0.0069

0.0090 0.0220 0.0080 0.0230 0.0570 0.0030 0.0200 0.0240 0.0300 0.0170 0.0090

− − − − − − − − − − −

Di-univalent electrolytes BaBr2 2.0 Ba(ClO4 )2 5.0 BaI2 2.0 CaBr2 6.0 6.0 CaCl2 Ca(ClO4 )2 6.0 CaI2 2.0 Ca(NO3 )2 6.0 CdBr2 4.0 CdCl2 6.0 CdI2 2.5 CoBr2 5.0 CoCl2 4.0 CoI2 6.0 Co(NO3 )2 5.0 CuCl2 6.0 6.0 Cu(NO3 )2 FeCl2 2.0 MgAc2 4.0 MgBr2 5.0

11094.48 −6669.99 115.7426 −95359.3 −27543.5 −27545.1 −13691.8 −6625.16 72636.26 68251.52 82358.18 −52552.5 −22132.8 −90257.5 −50823.7 12550.96 −32508.8 −2828.85 −20828 −52282

−12881.4 −8688.01 −12370.9 −7043.85 −7003.34 −7003.11 −11099.3 −3666 −32020.9 −30483 −35154.2 −8000.85 −6709.24 −10659.9 −2721.31 −12113.3 −2877.66 −9783.81 1231.443 −9413.14

0.0232 0.0419 0.0285 0.2790 0.1815 0.1815 0.0362 0.0336 0.1967 0.1512 0.3013 0.0784 0.0270 0.1721 0.0418 0.0357 0.0391 0.0239 0.0096 0.2111

0.0260 0.0720 0.0340 0.3510 0.2050 0.2720 0.0460 0.0600 0.2580 0.2140 0.3740 0.1410 0.0550 0.2420 0.1080 0.0380 0.1130 0.0290 0.0130 0.2410

0.020 0.021b 0.015 0.072 0.021 0.005 0.007 0.046 0.365 0.333 0.466 0.039 0.045 0.100 0.026 0.048 0.035 0.019 0.072 0.025

Electrolyte

Maximum molality

NaNO3 NaOH Na Pelargonate Na Propionate NH4 Cl NH4 NO3 RbAc RbBr RbCl RbI RbNO3 TiAc

6.0 6.0 2.5 3.0 6.0 6.0 3.5 5.0 5.0 5.0 4.5 6.0

Eme (J/mol)

E. Zhao et al. / Fluid Phase Equilibria 173 (2000) 161–175

171

Table 1 (Continued) Electrolyte

Maximum molality

Eme (J/mol)

Eem (J/mol)

ρ (Wilson)

ρ (NRTL)

ρ (NRTL–NRF)

MgCl2 Mg(ClO4 )2 MgI2 Mg(NO3 )2 MnCl2 NiCl2 Pb(ClO4 )2 Pb(NO3 )2 SrBr2 SrCl2 Sr(ClO4 )2 SrI2 Sr(NO3 )2 UO2 Cl2 UO2 (ClO4 )2 UO2 (NO3 )2 ZnCl2 Zn(ClO4 )2 Zn(NO3 )2

5.0 4.0 5.0 5.0 6.0 5.0 6.0 2.0 2.0 4.0 6.0 2.0 4.0 3.0 5.5 5.5 6.0 4.0 6.0

−43699.2 −59411.1 −112355 −73032.1 −1185.5 −58747.6 −61969.5 57151.77 −6352.86 −56192.5 −39994.6 −9064.63 10312.73 9365.232 −84975.7 12123.23 2885.854 −56890.2 −47502.5

−7218.98 −11296.9 −11026.4 −3021.82 −9519.21 −4009.6 −3054.93 −27037.3 −9559.72 −1890.45 −7737.33 −11144.9 −7648.35 −14118.8 −15534.9 −15619.4 −6234.68 −11547.7 −3842.78

0.1793 0.1804 0.2761 0.0483 0.0235 0.0314 0.0436 0.0165 0.0299 0.0438 0.0800 0.0375 0.0270 0.0343 0.4431 0.0415 0.1069 0.2099 0.0507

0.2020 0.2080 0.3160 0.1250 0.0470 0.0920 0.1470 0.0220 0.0360 0.0880 0.1680 0.0460 0.0290 0.0400 0.4470 0.0410 0.1190 0.2110 0.1480

0.018 0.026 0.046 0.022 0.067 0.052 0.028 0.064 0.013 0.020 0.042 0.006 0.041 0.024 0.029 0.094 0.029 0.019 0.021

67227.92 64041.20 64927.80 63276.19 63910.58 65400.98 63814.85 67014.83

−32761.87 −31329.43 −31490.84 −30754.75 −31025.21 −31680.08 −30915.47 −32314.27

0.0822 0.0514 0.0710 0.0414 0.0449 0.0659 0.0488 0.0889

0.0390 0.0360 0.0370 0.0310 0.0370 0.0380 0.0370 0.0500

0.078 0.074b 0.084 0.072 0.067 0.083 0.073 0.054

−61697.3 −10763 23168.75 19658.43 −13792.8 −7201.85 −13871.5 −11246.6 −14886.6 −13042.2 −29918.3

−9800.6 −10515.1 −19687.3 −18153.5 −10565.5 −11157 −9961.72 −10366.7 −11438.7 −10498.8 −8932.32

0.0802 0.0634 0.0458 0.0488 0.0690 0.0640 0.0830 0.0624 0.0586 0.0649 0.0646

0.1150 0.0840 0.0690 0.0540 0.0910 0.0820 0.0830 0.0820 0.0780 0.0870 0.0930

0.080 0.063 0.073 0.070 0.068 0.063 0.061 0.062 0.059 0.064 0.064

−10111.4 −18264.4

0.0648 0.1374

0.0750 0.1290

0.051 0.132

Di-divalent electrolytes BeSO4 4.0 MgSO4 3.5 MnSO4 4.0 NiSO4 2.5 CuSO4 1.4 ZnSO4 3.5 CdSO4 3.5 UO2 SO4 6.0 Tri-univalent electrolytes AlCl3 1.8 CeCl3 2.0 CrCl3 1.2 Cr(NO3 )3 1.4 EuCl3 2.0 LaCl3 2.0 NdCl3 2.0 2.0 PrCl3 ScCl3 1.8 SmCl3 2.0 YCl3 2.0 Tri-divalent electrolyte Al2 (SO4 )3 1.0 Cr2 (SO4 )3 1.2 a b

−12.6663 23030.58

Maximum molality is larger than the other two models. Maximum molality is smaller than the other two models.

172

E. Zhao et al. / Fluid Phase Equilibria 173 (2000) 161–175

Table 2 Comparison of the average standard deviations of the fit of the mean ionic activity coefficients of several electrolyte in aqueous solutions obtained from the new model presented in this study, NRTL–electrolytes model proposed by Chen et al. [4] and NRTL–NRF model proposed by Haghtalab and Vera [16] Electrolyte type

1:1

1:2

2:1

2:2

3:1

3:2

Number of systems This work NRTL NRTL–NRF

57 0.010 0.012 0.011

11 0.019 0.020 0.017

39 0.103 0.139 0.064

8 0.062 0.038 0.073

11 0.064 0.083 0.066

2 0.101 0.102 0.092

the average standard deviations obtained from a fixed value of the parameter C and its adjusted value showed a substantial improvement in the quality of the fit. For example, for uni-univalent electrolytes the adjustment of the parameter C will reduce the average standard deviation of the fit from 0.010 obtained by setting the value of the parameter C to 10–0.007. This provides the new model with more flexibility than the existing models in the literature to represent the thermodynamic properties of more complex electrolytes.

5. Conclusions In this work we developed a new activity model for electrolyte solutions. The activity coefficient of electrolytes in aqueous solutions is represented by a sum of the contributions of a long-range and a short-range interaction term. The long-range interactions were accounted for by the Pitzer–Debye–Hückel model. The short-range interactions were represented with a new expression based on the local composition concept. The main difference between this model and the extensions of the NRTL model to electrolytes is the assumption that the short-range energy parameter between species in a local cell has an enthalpic rather than commonly assumed Gibbs energy nature. As a result of this assumption, for systems containing only nonelectrolytes, our model simplifies to the three-parameter Wilson model. The new model contains two energy parameters, which should be evaluated from the experimental data and a coordination number that has been set to 10. The coordination number can also be treated as an adjustable parameter. It was found that although adjusting the coordination number increases the number of parameters in the model, it substantially increases the accuracy and capability of the model to represent aqueous solutions of complex electrolytes. The two-parameter version of the model, using a fixed value for the coordination number, was applied to several single electrolyte systems and it was shown that it could represent the experimental data with a better accuracy than the NRTL–electrolyte models [4,16]. List of symbols Notations Aφ Debye–Hückel constant in mole fraction scale C coordination number E dummy energy parameter G Gibbs Energy h enthalpy

E. Zhao et al. / Fluid Phase Equilibria 173 (2000) 161–175

H Ix K m MS n NP R T x X W Z

173

dummy parameter ionic strength in mole fraction scale dummy parameter molality molecular weight of solvent number of moles number of points universal gas constant absolute temperature mole fraction effective mole fraction potential of mean force charge number of ionic species

Greek symbols ε energy parameter γ activity coefficient ν stoichiometric number ρ closest approach parameter σ standard deviation ξ volume fraction Superscripts cal calculated exp experimental E excess LR long-range M mixing PDH Pitzer–Debye–Hückel SR short-range ∗ Henry’s convension ∞ infinite dilution Subscripts a anion c cation e electrolyte m solvent ref reference state Appendix A The activity coefficients of the species in solutions are assumed to be a sum of the contributions of a long-range electrostatic term obtained from the Pitzer–Debye–Hückel model [19], lnγ PDH , and a

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E. Zhao et al. / Fluid Phase Equilibria 173 (2000) 161–175

short-range interaction term, lnγ SR , as ln γi = ln γiPDH + ln γiSR

(A.1)

The expression for lnγ PDH is given as 

ln γiPDH

1000 =− Ms

0.5

 Aφ

Z 2 I 0.5 − 2Ix0.5 2Zi2 ln(1 + ρIx0.5 ) + i x ρ 1 + ρIx0.5

 (A.2)

The contribution of the short-range interaction term to the activity coefficient of the solvent, cation and anion can be obtained by combining Eq. (41) with Eq. (42). The expressions for the contribution of this term to the activity coefficient of the solvent m, cation c and anion a can be respectively written as • Activity coefficient of the solvent:   1 1 SR P P − ln γm = −Xm C Xm + k (Xck + Xak )Hek m Xm + k (Xck + Xak ) ! X 1 Hmek P P −C Xck − X H + X H X + m me a e e m k j j k j j X aj k ! X 1 Hmek P P − −C Xak Xm Hmek + j Xcj Hej ek X m + j X cj k P   Xm + k (Xck + Xak )Hek m P −C ln π (A.3) Xm + k (Xck + Xak ) • Activity coefficient of the cation in electrolyte k: ln γcSR k

Hek m 1 P P = − Xm C − Xm + j (Xcj + Xaj )Hej m Xm + j (Xcj + Xaj ) ! X Hek el 1 P P −C Xal − X H + X H X + m me c e e m k j j l j j X cj l # ! " P Xm Hmek + j Xaj Hej ek P − ln(Hmek ) − Hek m + 1 −C ln X m + j X aj

• Activity coefficient of the anion in electrolyte k: ln γaSR k

Hek m 1 P P = − Xm C − Xm + j (Xcj + Xaj )Hej m Xm + j (Xcj + Xaj ) ! X Hek el 1 P P − X cl C − X H + X H X + m me a e e m k j j l j j X aj l # ! " P Xm Hmek + j Xcj Hej ek P − ln(Hmek ) − Hek m + 1 −C ln X m + j X cj

!

(A.4)

!

(A.5)

E. Zhao et al. / Fluid Phase Equilibria 173 (2000) 161–175

175

where

  Eij Hij = exp − CRT

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]

K.S. Pitzer, J. Phys. Chem. 77 (1973) 268. K.S. Pitzer, G. Mayorga, J. Phys. Chem. 77 (1973) 2300. J.L. Cruz, H. Renon, AIChE J. 24 (1978) 817. C.-C. Chen, H.I. Britt, J.F. Boston, L.B. Evans, AIChE J. 28 (1982) 588. C.S. Lee, S.B. Park, Y.S. Shim, I & EC Res. 35 (1996) 4772. B. Sander, A. Fredenslund, P. Rasmusen, Chem. Eng. Sci. 41 (1986) 1171. E.A. Macedo, P. Skovborg, P. Rasmusen, Chem. Eng. Sci. 45 (1990) 875. H. Renon, J.M. Prausnitz, AIChE J. 14 (1968) 135. C.-C. Chen, L.B. Evans, AIChE J. 32 (1986) 444. G.M. Wilson, J. Am. Chem. Soc. 86 (1964) 127. P.J. Flory, Principles of Polymer Chemistry, Cornell University Press, Ithaca, New York, 1953. P.J. Flory, Discuss. Faraday Soc. 49 (1970) 7. M.L. Huggins, Macromolecules 4 (1971) 274. H. Renon, J.M. Prausnitz, AIChE J. 15 (1969) 785. F.X. Ball, W. Fürstm, H. Renon, AIChE J. 31 (1985) 392. A. Haghtalab, J.H. Vera, AIChE J. 34 (1988) 803. T.L. Hill, An Introduction to Statistical Thermodynamics, Dover, New York, 1986. E.A. Guggenheim, Proc. R. Soc., London Ser. A 183 (1945) 213. K.S. Pitzer, J. Am. Chem. Soc. 102 (1980) 2902. R.A. Robinson, R.H. Stokes, Electrolyte Solutions, Butterworths, London, 1970.

(A.6)