A square-well equation of state for aqueous strong electrolyte solutions

Fluid Phase Equilibria 285 (2009) 96–104

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A square-well equation of state for aqueous strong electrolyte solutions Ali Haghtalab ∗ , Seyed Hossein Mazloumi Department of Chemical Engineering, Tarbiat Modares University, P.O. Box: 14115-143, Tehran, Iran

a r t i c l e

i n f o

Article history: Received 26 April 2009 Received in revised form 30 June 2009 Accepted 20 July 2009 Available online 28 July 2009 Keywords: Equation of state Electrolyte Mean activity coefficient Osmotic coefficient Square-well

a b s t r a c t A simple and accurate electrolyte-equation of state is introduced for correlation of the mean activity coefficients of strong aqueous electrolyte solutions. The nonelectrolyte contribution of the new electrolyte-equation of state is based on the cubic square-well equation of state, SWEOS, that is composed of the van der Waals repulsive part of EOS and the attractive term is based on the square-well potential. The explicit version of the mean spherical approximation (MSA) is adopted for electrostatic contribution of electrolyte solution. The new electrolyte-equation of state, eEOS, has three adjustable parameters per each salt that are optimized by correlation of the mean activity coefficients of more than 130 binary aqueous electrolyte solutions at 25 ◦ C and 1 bar. Furthermore, the parameters are used for prediction of the osmotic coefficients and densities of several aqueous binary electrolyte solutions at 25 ◦ C. Also the mean activity and osmotic coefficient of a number of electrolytes have been correlated in temperature range of 0–100 ◦ C. Moreover, the ionic specific parameters are obtained for five binary aqueous electrolyte solutions by simultaneous fitting of mean activity, osmotic coefficient and apparent molal volume. The results demonstrate that the new eEOS can be applied for accurate representation of the activity and osmotic coefficients of the aqueous electrolyte solutions at wide range of the electrolyte concentration and temperature of binaries. © 2009 Elsevier B.V. All rights reserved.

1. Introduction Accurate representation of thermodynamic properties of electrolyte solution is vital in design of many processes in chemical industries such as natural gas treatment, extractive distillation, desalination, hydrometallurgy, biotechnology, etc. Generally, there are two approaches for thermodynamic modeling of electrolyte solution, one major group of models is based on excess Gibbs energy function and the other category is developed from a fundamental basis that is based on the residual Helmholtz energy equation. The models such as electrolyte virial Pitzer equation [1], electrolyteNRTL [2] and NRTL-NRF models [3] are some examples of the first group of the activity coefficient functions that are widely used in correlating and prediction of properties of electrolyte solutions. The advantages of these models are their relative simplicity in engineering design of chemical processes and their good accuracy for various electrolyte systems. However, the activity coefficient models have certain weaknesses such as lack of density calculation of the solution and direct calculation of the activity coefficient of electrolytes in terms of pressure. Based on residual Helmholtz energy, the electrolyte-equation of state is not subjected to the aforementioned weaknesses; however using electrolyte EOS is not straightforward

∗ Corresponding author. Tel.: +98 21 82883313; fax: +98 21 82883381. E-mail address: [email protected] (A. Haghtalab). 0378-3812/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.fluid.2009.07.018

in calculation of thermodynamic properties of electrolyte solutions as well as activity coefficient models. Using this approach, the pressure equation and the other thermodynamic properties can be obtained by appropriate differentiation of residual Helmholtz energy expression. Several electrolyte-equations of state, eEOS, have been proposed in the literature so far so that the most of them are known as extension of the common nonelectrolyte EOSs. Generally, the eEOS consists of three or four contributions that result from the Helmohltz energy. Similar to the cubic EOS the repulsive and attractive parts are expressed in terms of the hard sphere and dispersion forces. The other parts of eEOS are the discharging–charging and electrostatic terms that are represented by Born and mean spherical approximation (MSA) expressions, respectively. Planche and Renon [4] proposed an eEOS which contains a hard sphere and an attractive term for the interaction of all components and an implicit MSA [5,6] for long-range contribution. Jin and Donohue [7] developed an eEOS based on PACT EOS in which the similar expressions of PACT were adopted for ion–ion and ion–molecule interactions. In Fürst and Renon eEOS [8] a modified SRK EOS, a specific ionic attractive term and a simplified MSA term were combined. Wu and Prausnitz [9] used Peng–Robinson EOS plus association term for molecule–molecule interaction. They implemented the association and Born terms for ionic interactions. Galindo et al. [10] extended SAFT-VR model for aqueous electrolyte solution in which a simplified MSA term was used for long-range contribution. The eEOS was developed by Myers et al. [11] contains

A. Haghtalab, S.H. Mazloumi / Fluid Phase Equilibria 285 (2009) 96–104

a Peng–Robinson with translated volume parameter, an explicit modification of MSA and a Born term. Clarke and Bishnoi [12] applied a modified Peng–Robinson EOS and simplified MSA, in addition to a specific term for solvation effects. Cameretti et al. [13] used PC-SAFT with a Debye–Hückel term for long-range interactions. The eEOS of Tan et al. [14] includes the SAFT EOS plus a restricted primitive model of MSA for ionic interactions. Liu et al. [15] constructed an eEOS based on LJ-SAFT EOS. They used the low density expansion of nonprimitive MSA and association terms for electrolyte part. Lin et al. [16] developed four eEOS by combination of different short-range terms (SRK, PR and CPA) and various long-range terms (two versions of MSA, Debye–Hückel and Born equation). Kim and Lee [17] extended an eEOS based on a hydrogen-bonding nonrandom Lattice fluid model. The ionic interactions in their eEOS are taken into account by association contribution of solvent-cation and MSA term. Inchekel et al. [18] used CPA EOS with the simplified MSA and adopted two alternative terms for short-range interaction of ion–solvent, in the first term, the Born expression was applied using solvated diameter and in the second term the same expression was used as by Fürst and Renon [8]. As one can see, in the aforementioned eEOSs, the short-range interaction of species in electrolyte solution is taken into account by a nonelectrolyte EOS and often a version of MSA is used for long-range interaction. Also the Born equation is sometimes included for discharging–charging processes. Recently Haghtalab and Mazloumi [19] developed a new coordination number model for square-well fluids that is in good agreement with the simulation data. The coordination number model was then used to obtain two square-well equations of state, SWEOS, so that one of them is presented as a cubic EOS. The aim of this study is to extend the cubic SWEOS (CSW EOS) to aqueous electrolyte solutions and to test its capability for representation of the thermodynamic properties of electrolytes, especially the mean activity coefficient of fully ionized aqueous electrolyte solutions at 0–100 ◦ C.

2. Thermodynamic framework Wu and Prausnitz [9], Myers et al. [11], Kim and Lee [17] and Inchekel et al. [18] proposed a thermodynamic cycle for formation of the electrolyte solution from an ideal gas mixture and obtained an expression for residual Helmholtz energy of an electrolyte solution. A hypothetical ideal gas mixture of solvent molecules and ions species at temperature T and volume V is taken as reference mixture. The first step is to uncharge the ions so that this process is performed using Born equation. Following the discharging process, the ideal gas mixture containing the uncharged ions and molecule of solvents, isothermally compresses to liquid mixture at volume V. The change of state at this stage is expressed by a nonelectrolyte EOS so that in this work the cubic square-well equation of state, CSW-EOS, is adopted. The charging process of neutral ions is the next step that the Born equation with opposite sign is again used for this contribution. The final step, for achieving of a real electrolyte solution, is to take into account the long-range contribution of the electrostatic interaction of ions that is expressed by the explicit MSA model. So the molar residual Helmholtz energy of an electrolyte solution can be written by four above contributions as discharge

ares (T, v, xi ) = a − aid = (aBorn

charge

+ aBorn ) + aCSW + aMSA

(1)

where subscript “CSW” stands for the cubic square-well EOS. By differentiation of the Helmohltz energy respect to volume, the

97

pressure equation of state is expressed as



P =−

∂a ∂v





=− T,xi

∂aBorn ∂v





−

∂aCSW



∂v

T,xi

 − T,xi

∂aMSA ∂v

= PBorn + PCSW + PMSA

 T,xi

(2)

where aCSW = aCSW + aid . The fugacity coefficient and the residual chemical potential of each component in a mixture are calculated through the following relations: ϕi =

RT exp Pv

where res i RT



=

res i



(3)

RT

1 ∂(nt ares ) RT ∂ni

 +





 = T,V,nj = / i

1 ∂(nt aCSW ) RT ∂ni

1 ∂(nt aBorn ) RT ∂ni



 +

T,V,nj = / i

 T,V,nj = / i

1 ∂(nt aMSA ) RT ∂ni



T,V,nj = / i

i,Born i,CSW i,MSA = + + RT RT RT

(4)

So the expressions for each term in the total residual Helmohltz energy are proposed as follows. The Born equation [20] is used for the discharging–charging processes as discharge

aBorn = aBorn

charge

+ aBorn

=−

NA e2 4ε0



1−

1 D

  x Z2 i i

ions

(5)

i

so that the discharging and charging are carried out in vacuum and in a solvent, respectively. In the above relation “e” is the unit of elementary charge, ε0 is permittivity of the free space, D is the dielectric constant, Z is the charge number of ion. It should be noted that the hydrated diameter is used in the above equation if the solvation effect is taken into account. Using Eq. (5), the Born pressure EOS is obtained by proper differentiation of the Born residual Helmholtz energy respect to volume as NA e2 ∂D  ni Zi i 4ε0 D2 ∂V

2

PBorn =

(6)

ions

Similarly, the chemical potential expression which is used for calculation of fugacity coefficient, is obtained by differentiation of residual Helmholtz energy respect to number of mole, i,Born NA e2 = RT RT 4ε0



2 Z2 1 ∂D  ni Zi − 2 − i i i D ∂ni



1 1− D





(7)

ions

As it can be seen the Born model is contributed to both pressure and chemical potential equations, if one uses the Pottel’s model [21] for D; however if one uses a constant dielectric value and neglects the density dependence of solvent dielectric, ∂D/∂V = 0, the Born contribution to the pressure equation is vanished. Myers et al. [11] pointed out that both the Born and MSA contributions to the equation of state depend on the dielectric constant of the solvent of the electrolyte solution. Due to polarizability of the solvent molecules, the dielectric constant of the water decreases respect to presence of ions in the solution; however, accurate modeling of this effect with a simple analytical equation is not straightforward. Therefore, the dielectric constant of the pure solvent, water, has been used for all calculations here. It should be noted that independency of the dielectric constant of water on pressure was used in this work in a given temperature range [10], so we use the following density

98

A. Haghtalab, S.H. Mazloumi / Fluid Phase Equilibria 285 (2009) 96–104

independent relation, valid at 0–100 ◦ C, for water dielectric [22] as D = Ds = −19.291 +

29815 − 0.019678T + 1.3189 × 10−4 T 2 T

− 3.1144 × 10−7 T 3

i,CSW = ln RT



v − 4 v0

z − 2

(8)



v

+



2v

4 v0i z − i ln v − 4 v0 2

xm j j ij

Moreover, the linear dependence of Born Helmohltz energy model respect to mole fraction leads to no contribution of the Born effect on the chemical potential equation. Therefore, in this work the contributions of the Born term to the pressure and chemical potential equations are vanished. On the other hand, the summations in the Born term are over ions, thus, it has no effect on chemical potential of the solvent. The contribution of the short-range forces is expressed by using the cubic SWEOS [19] as aCSW = RT ln





v v − 4 v0

−

zRT ln 2



mv + v0 w mv + v0 (1 − m)



(9)

where the first term shows the van der Waals repulsive part and the second term denotes the attractive part based on square-well potential. R is gas √ constant, T is absolute temperature, v is molar volume and  = 2/6. For a liquid mixture, the following mixing rules are used as m=

 i

w=

 i

v0 =

xi xj mij

(10)

j

xi xj wij =

j



 i

 xi xj exp

ε  ij

kT

− mij

 (11)

j

xi v0i

(12)

i

z=



xi zi

−

NA i3

v0i = √ zi =

√ 4 2 3 ii − 1 3

ij =



aMSA = −

= 2 =

(20)

1+

3 

2

 (21)

1 [ 1 + 2 − 1] 2 e2 NA2 Dε0 RT v



(22)

xi Zi2

(23)

ions

2 3 RT 3NA



1+

3 

2



−

RT2

4NA 1 + 

(24)

(25)

As well as Born equation, the summations in the MSA term are over ions, so it has no effect on chemical potential of solvent molecule. Using the residual chemical potential, the unsymmetrical activity coefficient of ions in an electrolyte solution is calculated as i∗ =

ϕi (T, P, x) v(xe → 0) exp = v ϕi (T, P, xe → 0)



res − res (xe → 0) i i



RT

i = ion

s =

where ε is the square-well potential depth,  is the diameter of the particle,  is the potential range, z is maximum attainable coordination number, v0 is the closed packed volume, NA is Avogadro’s number and m is orientational parameter. In the above equations the summation is over all species. The known parameters for pure components are , and R that are obtained by optimization of the thermodynamic properties of the pure components; however for ions these parameters are calculated by using the data of the mean activity coefficient of binary electrolyte solutions. The pressure and chemical potential equations of CSW EOS are obtained as RT



(14)

(18)

i + j

v − 4 v0



; (26)

and solvent activity coefficient is obtained as

i i + j j

PCSW =

2 3 RT v 3NA

e2 NA Zi2 i,MSA

=− RT 4Dε0 RT 1 + 

(17)

+ (1 − m)v0i + 2v0

where is the

MSA screening parameter,  is the Debye screening

length and  = xi  i / xi where summation is over ions. Although the MSA is derived in McMillan–Mayer framework and it should be converted to the Lewis–Randall framework, nevertheless it has been shown that this incompatibility can be neglected [24], especially the present eEOS has three adjustable parameters so that this discrepancy may be compensated [11]. Similarly, the pressure and chemical potential equations are obtained as

(16)

εi εj

xw j j ij

mv + v0 (1 − m)

PMAS =

with the combining rules as εij =

xm j j ij

(13)

(15)

2



A few models such as Debye–Hückel and MSA have been used for long-range contribution of ions in the electrolyte solution. While the Debye–Hückel type models are frequently implemented in the activity coefficient models, a version of MSA theory is often used in the Helmohltz energy equation. In this work, the explicit simple version of MSA is adopted for long-range interaction of ions in the electrolyte solution [23] as

i

√ 4 2 3ij − 3 mij = √ 4 2( 3ij − 1)



mv + v0 w mv + v0 (1 − m)

mv + v0 w



2(v − v0 )

+ wv0i + 2v0



+

zRT (1 − m − w)mv0 2 (mv + v0 w)(mv + v0 (1 − m))

(19)

v(xs → 1) ϕs (T, P, x) exp = v ϕs (T, P, xs → 1)

 res − res (x → 1)  s s s RT

(27)

where v is the molar volume of liquid phase that is calculated by solving the pressure equation at given temperature and pressure. ϕ is fugacity coefficient, and subscripts e and s denote electrolyte and solvent, respectively. The mean activity coefficient on mole fraction basis can be obtained as ±x = ( + + − − )1/( + + − ) where

±x

(28)

is the mole fraction based of mean activity coefficient,

+ and − are stoichiometric numbers of cation and anion, respectively. The osmotic coefficient is calculated as  = −1000

ln(xs s ) ( + + − )Ms me

(29)

A. Haghtalab, S.H. Mazloumi / Fluid Phase Equilibria 285 (2009) 96–104

99

where me is molality of electrolyte and Ms is molar mass of solvent in g mol−1 . 3. Results and discussion 3.1. Pure water The three adjustable parameters of water were obtained by simultaneous fitting of saturated vapor pressure and liquid density of water in the temperature range of 283–373 K [25]. The values of 2.32839 Å, 528.61 K and 1.71391 were found for , ε/k and of water, respectively. The average absolute deviation of 1.0 and 0.14 were obtained for saturated vapor pressure and liquid density of pure water, respectively. 3.2. Aqueous binary electrolyte solution The new electrolyte-equation of state, eCSW, was used for correlation of the mean activity coefficient of 132 binary aqueous electrolyte solutions at 25 ◦ C and 1 bar [26]. We assumed that the values of the parameters of a cation and an anion in a binary electrolyte system are the same, i.e.  c =  a , εc = εa , c = a . Thus, the present model have three adjustable parameters per each electrolyte that have been determined through the following objective function: ı=

 | m,exp − m,cal | ±

± m,exp

±

Fig. 1. The correlations of the mean activity coefficients of three highly nonideal binary electrolytes at 25 ◦ C, the data: [26].

(30)

where ±m,cal =

±x 1 + 0.001( + + − )me Ms

(31)

m,exp

where ±m,cal , ± are the calculated and experimental mean activity coefficient based on molality scale, respectively, and the value of ±x is calculated using Eqs. (26) and (28). The results of the new eEOS is shown in Table 1 by presenting the optimized parameters of the eCSW EOS. Also Table 1 gives the comparison of the results of the new eEOS and Myere–Sandler–Wood EOS (MSW) [11]. As we mentioned in the introduction, the MSW EOS is the extension of the PR cubic EOS for correlation of the same activity coefficient data of binary electrolyte solutions. Also the MSW EOS used the same data source for the same number of electrolytes that we are used in this work. Based on overall AAD% given in Table 1, for the three electrolyte types, i.e., 1:1, 2:1 and 2:2 systems, the results of the present EOS are in better agreement with the experiment than the MWS EOS. The number of the 1:1, 2:1 and 2:2 electrolyte systems are 59, 45 and 8, respectively, while the 1:2 and 3:2 salt systems, in which the MSW EOS shows better accuracy, are 9 and 11, respectively. However, a peer assessment shows that about 32% of all presented cases, the both EOSs resulted in equal or similar quality. For at least 13% of cases both approaches were unsuccessful, even though the new EOS shows less deviation. So using the present EOS for the 2:1 electrolyte systems, the overall AAD% is less than the deviation obtained by MWS EOS. Also it comes out that in 22% of electrolytes the new eEOS was more accurate, whereas in about 33% of cases the MSW EOS was more successful. Fig. 1 shows the experimental and calculated mean activity coefficients of the three electrolyte solutions with strong positive and negative deviation from ideality. As one can see the results of the new eCSW are in very good agreement with experimental data. Also Figs. 2 and 3 show the predictions of the osmotic coefficients and densities of a few electrolyte solutions, respectively. As one can see the predictions of the osmotic coefficients of the electrolytes have been carried out with very good accuracy, however the predicted densities is rather good because the density data are not considered

Fig. 2. The predictions of the osmotic coefficients of five electrolytes at 25 ◦ C, the data: [26].

Fig. 3. The predictions of solution densities of four electrolytes at 25 ◦ C, the data: [27].

100

A. Haghtalab, S.H. Mazloumi / Fluid Phase Equilibria 285 (2009) 96–104

Table 1 The optimized parameters of the new eCSW and comparisons of the results with MSW EOS [10], the data: [26]. Electrolyte

Max. me

 (Å)

ε/k (K)



AAD%a This work

(1:1) AgNO3 CsAc CsBr CsCl CsI CsNO3 CsOH HBr HCl HClO4 HI HNO3 KAc KBrO3 KCl KClO3 KCNS KF KH adipate KH malonate KH succinate KTol KH2 PO4 KI KNO3 KOH KBr LiAc LiBr LiCl LiClO4 LiI LiNO3 LiOH LiTol NaAc NaBr NaBrO3 NaCl NaClO3 NaClO4 NaCNS NaF Na formate NaH malonate Na propionate NaH succinate NaTol NaI NaNO3 NaOH NH4 Cl NH4 NO3 RbAc RbBr RbCl RbI RbNO3 TlAc

6 3.5 5 6 3 1.4 0.9 3 6 6 3 3 3.5 0.5 4.5 0.7 5 4 1 5 4.5 3.5 1.8 4.5 3.5 6 5.5 4 6 6 4 3 6 4 4.5 3.5 4 2.5 6 3.5 6 4 1 3.5 5 3 5 4 3.5 6 6 6 6 3.5 5 5 5 4.5 6

3.0219 4.3340 3.0739 3.0262 2.9756 2.9240 4.9733 4.4439 4.1998 4.4646 4.3075 4.3056 4.2450 4.0530 3.9064 3.9640 3.8662 3.9742 3.8885 3.7771 3.8958 4.4482 3.3176 4.0357 3.0661 3.9444 3.8794 3.9896 4.3205 4.3000 4.4966 4.7258 4.0360 1.2737 4.2374 4.2016 4.1494 3.9741 4.0872 4.1100 4.0065 4.2753 4.2132 4.1063 3.6163 4.2203 3.7595 4.2194 4.2734 3.7041 4.2322 3.7451 3.0838 4.2797 3.6474 3.6697 3.6100 2.7494 2.8341

851.78 1084.52 845.86 836.43 648.89 890.92 1303.14 1147.12 1078.53 1151.55 1083.16 1091.35 1062.40 885.49 982.63 842.28 949.98 1019.44 902.30 955.52 999.58 1119.28 989.47 999.92 828.08 1029.17 965.81 997.00 1120.20 1103.98 1139.35 1201.68 1005.14 530.21 1062.82 1047.36 1054.91 1022.55 1036.54 1034.37 994.06 1064.75 1097.52 1015.89 894.36 1048.94 956.08 1046.56 1085.50 913.70 1082.75 919.87 711.64 1074.04 915.82 930.24 915.08 812.70 645.81

1.7672 1.6114 1.6413 1.6306 1.7285 1.7251 1.4697 1.5279 1.5553 1.5474 1.5324 1.6236 1.6129 1.9929 1.6834 1.9925 1.7107 1.6352 1.7644 1.7279 1.6937 1.7476 1.6755 1.6832 1.7785 1.5565 1.6841 1.6420 1.5450 1.5771 1.5633 1.5527 1.6192 1.1715 1.6952 1.6282 1.6303 1.7076 1.6485 1.7006 1.6892 1.6580 1.6523 1.6821 1.7140 1.6083 1.6819 1.7472 1.6131 1.7355 1.6292 1.6969 1.7911 1.6071 1.6889 1.6776 1.6810 1.7643 1.7306

Overall (1:2) Cs2 SO4 K2 SO4 K2 CrO4 Li2 SO4 Na2 CrO4 Na2 SO4 Na2 S2 O3 (NH4 )2 SO4 Rb2 SO4 Overall

1.8 0.7 3.5 3 4 4 3.5 4 1.8

4.2861 3.5727 4.1774 4.2435 4.4722 4.0963 4.2739 3.3673 4.3085

1111.26 707.22 1068.75 1083.98 1144.58 1066.26 1101.98 909.97 1121.24

1.6792 1.9629 1.7003 1.6671 1.6804 1.7114 1.6881 1.6963 1.6846

MSW

0.06 0.20 0.21 0.16 0.14 0.15 0.27 0.11 0.13 0.34 0.72 0.13 0.22 0.03 0.14 0.04 0.11 0.08 0.19 0.56 0.23 0.16 0.06 0.27 0.11 0.36 0.18 0.25 0.35 0.12 0.39 0.14 0.28 0.29 0.29 0.11 0.11 0.08 0.17 0.21 0.21 0.13 0.02 0.06 0.29 0.28 0.19 0.16 0.06 0.14 0.43 0.26 0.16 0.16 0.07 0.07 0.09 0.16 0.20

0.2 0.65 0.31 0.28 0.17 0.15 0.46 0.09 0.63 0.2 2.61 0.06 0.61 0.08 0.08 0.08 0.13 0.08 0.09 0.52 0.22 0.35 0.08 0.2 0.13 0.6 0.09 0.3 0.37 0.25 4.26 1.29 1.55 1.01 0.58 0.61 0.09 0.08 0.08 0.09 0.07 0.36 0.02 0.31 0.26 0.76 0.2 0.44 0.19 0.11 0.67 0.29 0.16 0.83 0.07 0.07 0.12 0.19 0.23

0.19

0.42

0.26 0.16 0.38 0.46 0.24 0.18 0.31 0.28 0.19

0.22 0.28 0.25 0.27 0.41 0.16 0.17 0.25 0.15

0.27

0.24

A. Haghtalab, S.H. Mazloumi / Fluid Phase Equilibria 285 (2009) 96–104

101

Table 1 (Continued ) Electrolyte

Max. me

(2:1) BaBr2 BaCl2 Ba(ClO4 )2 BaI2 Ba(NO3 )2 BaAc2 CaBr2 CaCl2 Ca(ClO4 )2 CaI2 Ca(NO3 )2 Cd(NO3 )2 CdCl2 CdBr2 CdI2 CoBr2 CoCl2 CoI2 Co(NO3 )2 CuCl2 Cu(NO3 )2 FeCl2 MgAc2 MgBr2 MgCl2 MgI2 Mg(ClO4 )2 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 ZnBr2 ZnCl2 ZnI2 Zn(ClO4 )2 Zn(NO3 )2

2 1.8 5 2 0.4 3.5 6 6 6 2 6 2.5 6 4 2.5 5 4 2 5 6 6 2

4 5 5 5 4 5 6 5 6 2 2 4 6 2 4 3 5.5 5.5 6 6 6 4 6

 (Å)

4.6165 4.6106 4.5604 4.7300 3.7609 3.9724 4.8076 4.5911 4.6765 4.8023 4.2229 4.5491 1.1140 0.9019 0.8865 4.3967 4.3229 4.5733 4.4436 4.2024 4.4019 4.6388 3.7959 4.7571 4.7079 4.8971 4.9289 4.5548 3.9176 4.2862 4.4255 2.9152 4.7587 4.5517 4.3834 4.8754 4.3056 4.4691 4.9372 4.0249 4.7848 4.7122 4.4171 4.8723 4.4794

ε/k (K)

1171.19 1168.88 1125.40 1203.11 1029.29 661.93 1233.42 1177.12 1193.65 1227.82 1043.31 1125.36 948.85 1077.15 1296.93 1129.72 1088.76 1175.20 1124.34 972.23 1106.16 1187.31 956.62 1222.38 1210.40 1255.76 1258.57 1149.87 898.33 1092.98 1119.09 879.15 1219.37 1161.66 1118.55 1244.53 1064.85 1090.72 1272.02 754.11 1121.91 1150.25 693.90 1255.04 1125.01



1.5810 1.6157 1.6138 1.5154 1.7011 1.9992 1.5095 1.5445 1.4724 1.4765 1.6745 1.6441 1.5676 1.4810 1.2754 1.4517 1.5420 1.4089 1.5670 1.6951 1.5952 1.5467 1.6211 1.4678 1.5120 1.4316 1.4325 1.5608 1.5581 1.5156 1.5530 1.7135 1.5332 1.5688 1.4830 1.4967 1.7267 1.5794 1.3271 1.4183 1.7263 1.7274 1.9009 1.4093 1.5660

Overall (2:2) BeSO4 MgSO4 MnSO4 NiSO4 CuSO4 ZnSO4 CdSO4 UO2 SO4

4 3.5 4 2.5 1.4 3.5 3.5 6

3.5093 3.6154 3.6471 3.4911 3.2180 3.6433 3.5156 3.1754

1056.76 1069.34 1091.34 1053.36 546.16 1115.33 1057.42 896.78

1.3993 1.4695 1.4936 1.4889 1.1282 1.4570 1.5073 1.4415

Overall (3:1) AlCl3 CeCl3 CrCl3 Cr(NO3 )3 EuCl3 LaCl3 NdCl3 PrCl3 ScCl3 SmCl3 YCl3

1.8 2 1.2 1.4 2 2 2 2 1.8 2 2

Overall a

AAD% =

100 np

 | m,cal − m,exp | ±,i

i

±,i m,exp ±,i

5.0359 4.8238 5.0400 4.9158 4.8319 4.8030 4.8009 4.8050 4.7986 4.8166 4.8399

1293.48 1226.97 1291.71 1248.44 1229.61 1220.13 1224.39 1223.72 1225.47 1226.21 1237.99

1.4364 1.5299 1.4584 1.5122 1.5167 1.5322 1.5224 1.5285 1.4800 1.5196 1.4988

AAD%a This work

MSW

0.43 0.35 1.98 0.20 0.09 0.70 3.08 0.58 1.33 0.74 0.50 0.64 13.18 19.86 38.49 1.83 1.67 5.95 0.74 0.75 0.95 0.53 0.30 0.94 0.72 1.74 0.71 0.38 1.92 2.26 1.30 0.18 0.44 0.57 1.33 0.61 0.50 0.73 1.44 4.15 0.98 1.19 5.42 0.58 0.91

0.16 0.17 4.75 0.3 0.11 4.21 2.11 0.71 3.9 0.18 0.21 0.52 21.28 26.53 45.56 4.7 2.65 18.92 0.73 4.59 1.71 0.22 0.96 0.27 0.39 0.5 0.48 0.34 6.8 3.89 2.68 0.19 0.16 0.3 6.45 0.23 0.23 2.24 1.69 15.65 4.24 1.18 14.06 0.38 2.15

2.75

4.66

2.07 2.03 1.85 1.54 1.20 2.13 1.51 1.05

1.86 2.17 1.78 1.76 1.56 2.05 1.53 1.71

1.67

1.80

0.58 0.48 1.00 0.63 0.70 0.76 0.73 0.91 0.84 0.79 0.54

0.61 0.44 0.49 0.21 0.27 0.16 0.22 0.32 0.29 0.23 0.29

0.72

0.32

102

A. Haghtalab, S.H. Mazloumi / Fluid Phase Equilibria 285 (2009) 96–104

Table 2 The temperature and molality ranges of the various electrolytes used in this study. Electrolyte

Property

NaCl NaBr NaNO3 KCl Li2 SO4 Na2 SO4 K2 SO4 Cs2 SO4

,  ,    ,  ,  ,  , 

Molality (mol/kg) 0.1–6 0.1–8 0.1–10 0.1–6 0.1–3 0.1–3 0.1–2 0.1–6

Temp. (◦ C)

Ref.

0–100 0–100 0–100 0–100 0–100 0–100 0–100 25–100

[28] [29] [30] [31] [32] [32] [32] [32]

in the optimization. So as mentioned by Myers et al. [11], it is often difficult to represent the density data with a cubic EOS well. To extend the present model to the other temperatures, nine aqueous electrolyte solutions have been selected that are given in Table 2. To cover entire range of temperature the following temperature dependent relation has been adopted for the electrolyte parameters,



pa = b0 + b1 (T − 298.15) + b2 T ln

 298.15  T



+ (T − 298.15) ;

pa = ; ε/k;

Fig. 4. The correlations of the mean activity coefficients of aqueous NaCl solution at different temperature, the data: [28].

(32)

As shown in Table 3, the coefficients of Eq. (32) were evaluated by fitting the experimental mean activity coefficient data through the objective function, Eq. (30). In the case of NaCl, the value of coefficient b0 in Eq. (32) is the same as given in Table 1, because the experimental data are taken from the two different references [26,28] are almost entirely the same at 25 ◦ C, however, in the other cases, for example NaBr, the maximum molality is different from the two references so that the fitted values of b0 is given in Tables 1 and 2 is slightly dissimilar. Also Table 3 shows the AAD% for nine electrolyte solutions. So as one can see the results of the present model are in very good agreement with the experimental data. Fig. 4 shows the comparison of correlated and experimental mean activity coefficient of aqueous NaCl at three different temperatures and 1 bar. It has been seen that the eCSW EOS represents the experimental data accurately. The performance of the new eEOS at high pressure is demonstrated by prediction of the osmotic coefficient of aqueous NaCl at three different temperatures and 200 bars using the parameters are given in Table 3. So as Fig. 5 shows the predicted results are in very good agreement with experiment. One may consider that there are two types of the interaction parameters that have been used for electrolyte solutions so that one group of models is based on electrolyte parameters and the other group uses the specific ionic parameters. At this work, for the new EOS we used the electrolyte parameter for each electrolyte as given in Table 1. One should be noted that using the specific ionic parameter for the whole electrolytes of any type is possible but it may imply a lot of constraints and tedious to optimize. However to test the potential of the new eEOS in terms of the ionic

Fig. 5. The prediction of the osmotic coefficients of aqueous NaCl solution at different temperature, the data: [28].

parameters, five electrolytes: LiCl, NaCl, KCl, RbCl and CsCl have been studied. So the experimental data of mean activity coefficient, osmotic coefficient [26] and apparent molal volume (AMV) [33] of these five electrolytes have been simultaneously adopted in optimization with equal weights to obtain the specific ionic parameters that are given in Table 4 and the AAD% in Table 5. Since the apparent molal volume of an electrolyte is much sensitive to molality variation respect to density, therefore it is often difficult to model

Table 3 The coefficients of temperature dependent parameters of the present model and the percent of absolute average deviation from experimental data for different aqueous electrolytes. Electrolyte

NaCl NaBr NaNO3 KCl Li2 SO4 Na2 SO4 K2 SO4 Cs2 SO4

 (Å)

ε/k (K)



AAD%

b0

1000b1

1000b2

b0

b1

b2

b0

1000b1

1000b2





4.0872 4.1177 3.8506 3.5795 4.0577 3.9582 3.6631 3.7727

−2.9692 −2.6542 5.5432 5.6135 −3.0789 −2.9817 −0.9678 2.6646

−9.4815 −6.4846 15.3782 16.3619 0.9064 1.5107 1.9706 50.4205

1036.54 1039.84 503.95 483.51 1039.95 1032.93 967.24 941.90

0.49619 0.63166 −0.02114 −0.01817 0.47327 −0.72614 −0.96561 1.23824

−1.56487 −0.37293 0.41082 0.49112 0.79227 −2.10092 −2.16853 3.51524

1.6485 1.6334 2.7090 2.5803 1.6647 1.7132 1.7050 1.7180

−1.42534 −1.64022 0.85405 0.92989 −0.89139 −0.23838 0.09836 −0.06532

−3.8142 −3.9783 −3.5159 −4.2335 −4.0096 −0.5792 −1.5047 13.9904

0.18 0.67 – – 0.60 0.72 0.68 1.54

0.14 0.48 0.54 0.38 0.48 1.00 0.86 1.43

A. Haghtalab, S.H. Mazloumi / Fluid Phase Equilibria 285 (2009) 96–104 Table 4 The specific ionic parameters optimized by simultaneous regression of mean activity coefficient, osmotic coefficient and apparent molal volume of the corresponding binary aqueous electrolytes.

T

v V vdW

Ion

 ion (Å)

εion /k (K)

ion

v0

Li+ Na+ K+ Rb+ Cs+ Cl−

1.8548 1.8101 2.2871 2.5736 2.8785 1.7959

1010.53 820.04 646.41 742.54 923.25 501.54

1.2757 1.3648 1.2094 1.6908 1.9456 1.1356

xi z Zi w

Table 5 The percent of absolute average deviation of mean activity, osmotic coefficient and apparent molal volume (AMV) [26,33] for five aqueous electrolytes using specific ionic parameters. Electrolyte

Max. me ( , )

Max. me (AMV)

LiCl NaCl KCl RbCl CsCl

6 6 4.5 5 6

3.5 3.5 2.5 2 1.6

AAD%



AMV

4.03 3.70 2.46 2.85 1.27

2.09 2.08 1.47 1.42 0.63

2.58 4.02 2.06 1.58 1.30

AMV data rather than density data [16]. So having this fact, one can conclude that the results given in Table 5 seems to be satisfactory. On the other hand, based on optimization of the mean activity coefficient only, i.e. Eq. (30), the AAD% of less than 0.3% have been obtained for activity and osmotic coefficient for these five electrolyte using ionic specific parameters, however the parameters aren’t reported here. 4. Conclusion A new electrolyte-equation of state was developed. The cubic nonelectrolyte part of the new eEOS was combination of vdW repulsive equation and the attractive term which is based on the square-well potential function. The explicit version of MSA theory was used for long-range contribution of electrolyte solution. The mean activity coefficients of 132 aqueous electrolyte solutions have been successfully correlated by the new eEOS at 25 ◦ C. Also the osmotic coefficients and solution densities of some electrolyte solutions were predicted. In addition, for nine aqueous binary electrolyte, the new eCSW EOS was extended to higher temperatures up to 100 ◦ C and the results were obtained with good accuracy. Finally, the satisfactory results were presented for simultaneous correlation of mean activity coefficient, osmotic coefficient and apparent molal volume for five aqueous electrolyte solutions via ionic specific parameters. The results demonstrated that the new eEOS is a powerful tool for representation of the activity and osmotic coefficients of aqueous electrolyte solutions. List of symbols a molar Helmholtz free energy (J mol−1 ) bi coefficients in Eq. (32) D dielectric constant of solution dielectric constant of solvent Ds e electronic charge (1.60219 × 10−19 C) k Boltzmann’s constant (1.38066 × 10−23 J K−1 ) m orientational parameter defined in Eq. (14) me molality (mol/kg) Ms molecular weight of solvent (g mol−1 ) n total number of mole (mol) Avogadro’s number (6.02205 × 10−23 mol−1 ) NA P pressure (Pa) R gas constant (8.314 J mol−1 K−1 )

103

temperature (K) molar volume (m3 mol−1 ) volume (m3 ) van der Waals close-packed volume (m3 mol−1 ) defined in Eq. (15) mole fraction of component i maximum coordination number charge number of ionic species i a function of temperature defined in Eq. (11)

Greek letters

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