Modification and application of a non-primitive mean spherical approximation model for simple aqueous electrolyte solutions

Fluid Phase Equilibria 209 (2003) 13–27

Modification and application of a non-primitive mean spherical approximation model for simple aqueous electrolyte solutions M. Lotfikian, H. Modarress∗ Department of Applied Chemistry, Faculty of Chemical Engineering, Amirkabir University of Technology, 424 Hafez Avenue, Tehran 15875, Iran Received 28 June 2002; received in revised form 18 February 2003; accepted 18 February 2003

Abstract In this study, the non-primitive mean spherical approximation (NPMSA) model has been modified and an equation is obtained for calculating the cationic diameter for single electrolyte solutions at various concentrations. Furthermore, based on this modification of the NPMSA model an equation for the Helmholtz energy is derived and the thermodynamic properties are calculated at various ionic strengths and dipole diameters. The results show that the pair potential and electrostatic forces play the main role in the obtained thermodynamic properties. Also, the calculated thermodynamic properties by the NPMSA model are compared with results from other models such as primitive mean spherical approximation (PMSA) and restricted mean spherical approximation (RMSA). The results indicate that the NPMSA model underestimates the thermodynamic properties due to the solvent effects in this model. © 2003 Elsevier Science B.V. All rights reserved. Keywords: Mean spherical approximation (MSA); Non-primitive model; Mean ionic activity coefficient; Thermodynamic properties; Electrolyte solutions; Ionic diameter; Statistical mechanics

1. Introduction Studying and representing the thermodynamic properties of real ionic solutions is essential for many technological and scientific applications. In recent years, increasing attention is paid to the statistical mechanical approaches such as molecular simulation, perturbation theory and integral equation theory for studying electrolyte solutions [1]. Based on the statistical mechanics, the Ornstein–Zernike (O–Z) integral equation can be solved under some simplified conditions by the mean spherical approximation (MSA) [2,3]. Although it is not as ∗

Corresponding author. Tel.: +98-911-222-0454; fax: +98-021-640-5847. E-mail address: [email protected] (H. Modarress). 0378-3812/03/$ – see front matter © 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0378-3812(03)00073-6

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M. Lotfikian, H. Modarress / Fluid Phase Equilibria 209 (2003) 13–27

accurate as the hypernetted chain equation (HNC) [4–9] or even the Percus–Yevick approximation, it is an analytical theory for modelling of electrolytes and is very useful in representing the thermodynamic properties of ionic solutions over a wide range of concentrations. Just as the classical Debye–Hueckel (DH) theory [10], which is used in developing some semi-empirical thermodynamic models of electrolyte solutions such as Pitzer equation [11], the MSA model is obtained from the linearized Poisson–Boltzmann equation, but instead of ignoring the exclusion volumes of the ions in the screening cloud, it takes exactly the ions as hard-spheres with charge, into account. Consequently, the restricted mean spherical approximation (RMSA) and primitive mean spherical approximation (PMSA) is more accurate for dense systems, in particular, in the unphysical asymptotic high-density limit in which the molecules fill all space. A remarkable property of the restricted and primitive MSA is that, just as the DH theory, it provides a one-parameter description of the thermodynamic excess properties of an arbitrary complex mixture of ions, ranging from a solution of simple ions to colloidal solutions and to molten salts. Here, ion–ion interaction parameter or screening parameter for ions and solvent are taken into account, and as a result the solution behaviour is considered as a continuum dielectric. In restricted MSA the ionic sizes should be equal to each other but in primitive MSA they could be different. In the non-primitive mean spherical approximation (NPMSA) that was investigated by Blum and co-workers [12–18], all the properties of a mixture of arbitrary size and charged ionic hard-spheres in a dipolar solvent are expressible in terms of three explicit parameters; a screening or ion–ion interaction parameter Γ , an ion–dipole interaction parameter B10 and a dipole–dipole interaction parameter b2 , which are given by the solution of a set of algebraic equations. The resulting equations in the NPMSA can be solved by numerical methods proposed by Lee [18], where the ionic diameters can be calculated by iteration method and by considering the Pauling diameters as the initial values. In this work by using a novel approach to the NPMSA model of aqueous electrolyte solutions, the parameters of this model are modified. A new equation is suggested for calculating the cationic diameter as a function of ionic strength of aqueous electrolyte solutions. The Helmholtz free energy of the electrolyte solutions is calculated by driving a simple equation based on modification of the NPMSA equations. The other thermodynamic properties are also calculated and the results are compared with those calculated by application of similar models. The differences in the calculated results are justified by considering the inherent assumptions in each model.

2. NPMSA equations For an electrolyte solution containing ions and dipoles mixture, the Ornestein–Zernike equation can be expressed as follows [17,18]:  n
hij (r12 ) = Cij (r12 ) + ρk Cik (r13 )hkj (r32 ) dr3 (1) k=1

where hij (r12 ) and Cij (r12 ) are, respectively, the total and direct correlation functions for particles i and j at positions r1 and r2 and n denotes the number dipole molecules. Cik (r13 ) and hkj (r32 ) have the same meaning for particles i and k at positions r1 , r2 and r3 and ρk denotes the number density of particle k.

M. Lotfikian, H. Modarress / Fluid Phase Equilibria 209 (2003) 13–27

15

In order to solve the O–Z equation, a so-called closure relation between hij (r12 ) and Cij (r12 ) is required. The MSA, is based on the following closure relation of the O–Z equation hij (r) = −1, Cij (r) = −βUij (r),

r < δij r > δij

(2)

where δij = (δi + δj )/2, δi and δj are the diameters of particles i and j, respectively, and Uij (r) represents the pair potential between i and j. For primitive and non-primitive models, analytic solution had been derived by Blum–Wei [19,20] and Waiseman–Lebowitz [21]. The analytical solution of Blum and Wei is for equal ionic sizes or arbitrary sizes of ions and solvents of point dipoles for an aqueous electrolyte solution system. In what follows we present the basic equations of NPMSA. We assume a system consisting of n − 1 charged hard-spheres of diameter δi , number density ρi , and charge Zi e, where e is the elementary charge and Zi the valence of the ion i. In this system, the solvent has a hard-core diameter δn , number density ρn and dipole moment µ. Because of the long range of the ion–ion, ion–dipole and dipole–dipole interactions, the direct correlation function is also long ranged, and we may write Cijmnl (r) = Cij(0)mnl (r) − βUijmnl (r)

(3)

where Uijmnl (r) is the radial part of the electrostatic interaction or pair potential, Cij(0)mnl (r) is the short ranged direct correlation function and superscript (0) denote the reference state. Clearly, Uij000 (r)

Zi Zj e2 = , r

Cij(0)mnl (r) r

,

Uin110 (r)

Zi eµ = 2 , r

 112 Unn (r)

=−

10 µ2 3 r3

(4)

In Eqs. (3) and (4), m, n and l are Wigner 3-j symbols and all refer to the perturbed states. By performing the Baxter factorisation [22] and use the three characteristic parameters Γ , B10 and b2 which are related to the ion–ion, ion–dipole and dipole–dipole terms, respectively, the equations of NPMSA can be solved and the thermodynamic properties can be expressed analytically in terms of these parameters. The three implicit parameters (Γ , B10 and b2 ) with a set of three non-linear boundary condition equations are derived by Blum et al. [23] in the following form. In addition, these equations and their parameters have been used to calculate the activity coefficients of electrolyte solutions by Li and co-workers [24,25]. 

0 2 i ρi (ai )



−

+ ρn (an1 )2 = d02

0 10 i ρi ai kni

11 + an1 (1 − ρn knn ) = d0 d2  10 2 11 2 (1 − ρn knn ) + ρn i ρi (kni ) = y12 + ρn d22

The relevant parameters are defined by the following equations:

(5)

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M. Lotfikian, H. Modarress / Fluid Phase Equilibria 209 (2003) 13–27

d02 =

4πβe2 , 4πε0

β3 = 1 +

b2 , 3

n−1

w1 =

d22 =

i=1

4πβµ2 12πε0

β6 = 1 −

b2 , 6

λ=

β3 , β6

y1 =

4 β6 (1 + λ)2

ρi Zi2 [β6 (δn + λδi )(1 + Γδi )]

n−1
1 ρi Zi2 δ2i w2 = ρn δ2n B10 2 [2β6 (δn + λδi )(1 + Γδi )]2 i=1  [−(w1 /2) + (w1 /2)2 + 2B10 w2 /β62 ] νη = w2

νη ρn δ2n δ2i B10 8β6 (δn + λδi )

!Γi = Γis =

[(1 + δi Γ − !Γi )D − 1] δi

DiF =

Zi β6 [2(1 + δi Γ − !Γi )]

D = 1 + νη2 ρn δ2n Ω10 = νη



β6 Γis DiF Dac

10 −kni =

,

Dac =

n−1


ρi (DiF )2

i=1

an1 =

Dβ6 [(δn B10 /2) + (Ω10 λ/Dβ6 )] 2Dac

δ2n DiF [νη /(δn + λδi ) + (Ω10 Γis /Dac )] 2Dβ62 + δ3n B10 ai0 /12β6

11 1 − ρn knn =

Ni =

ρi δ2i (DiF )2 i [2β6 (δn + λδi )]2

 ρi δi (DiF )2 i δn + λδi

νη DiF mi = , (δn + λδi ) a10 =

(6)

[λ + (ρn δ2n Ω10 an1 /2β62 )] Dβ6 + ρn δ3n B10 an1 /12β6

2DiF [1 + (ρn δ3n B10 νη δi /24(δn + λδi ))] β6 δi − Zi /δi

The set of Eq. (5) can solve numerically by Newton–Raphson method. The ranges of Γ , B10 and b2 suggested by Lvov and Wood [26] are adopted and their initial values are 0.45, 0.02 and 1.5, respectively.

M. Lotfikian, H. Modarress / Fluid Phase Equilibria 209 (2003) 13–27

The objective function (OBJ) in this calculation is defined as  OBJ = 13 F1 + F2 + F3

17

(7)

where F1 , F2 and F3 are given by the following equation set functions

10 11 F1 = ρi (ai0 )2 + ρn (an1 )2 − d02 , F2 = − ρi ai0 kni + an1 (1 − ρn knn ) − d0 d2 , i

i


11 2 10 2 ) + ρn ρi (kni ) − y12 − ρn d22 F3 = (1 − ρn knn i

In our calculations, the ion–ion coupling parameter, d0 and the dipole–dipole strength parameter, d2 are modified. The obtained parameters (Γ , B10 and b2 ) are used to calculate the activity coefficient and thermodynamic parameters. Then correlating the experimental and theoretical activity coefficients the ionic diameters in various concentrations are calculated. 3. Thermodynamics of the NPMSA model In the NPMSA model, the thermodynamic properties can be expressed in terms of three interaction parameters (Γ , B10 and b2 ) [19,20]. In this model, the internal energy change !E over a hard-sphere reference system is  2  d0 i ρi Zi Ni − 2d0 d2 ρn B10 − 2d22 ρn b2 /δ3n β !E = (8) V 4π The results are compared with the primitive and the restricted MSA and the RMSA [18]. In NPMSA, the pressure can be obtained as βEMSA − βAMSA V The Gibbs energy change of an ion placed in a continuum dielectric is [20] 2 2  1 Zi e 1− !G = − δi ε βP MSA =

(9)

(10)

The ionic chemical potential can be derived in the following form: Zi (d02 Ni − d0 d2 ρn mi ) 4π For dipole chemical potential we have βµMSA = i

[−d0 d2 B10 − 2d22 b2 /δ3n ] 4π The chemical potentials of the hard-sphere mixture is given by the MCSL equation [27] βµn =

πP hs δ3i β 3δi (ξ2 + ξ1 δi ) 9ξ22 δ2i βµhs = −ln ∆ + + + i 6 ∆ 2∆2  2



2

ξ2 δi ξ3 (ξ3 /∆) ξ3 (1 + ∆) ξ2 δi 3 +3 ln ∆ + − 2 ln ∆ + − ξ3 ∆ ξ3 2 ∆

(11)

(12)

(13)

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M. Lotfikian, H. Modarress / Fluid Phase Equilibria 209 (2003) 13–27

with πP hs β ξ0 3ξ1 ξ2 (3 − ξ3 )ξ23 = + + , 6 ∆ ∆2 ∆3

n

ξl =

π
l ρi δ , l = 0, 1, 2, 3 6 i=1 i

∆ = 1 − ξ3

(14)

The excess chemical potential of ion i for an electrolyte system is MSA + βµhs βµex i = βµi i

(15)

By considering the infinite dilution as the reference state of the ionic activity coefficient, we have [24] ex ln fi = β[µex i (xi ) − µi (xi → 0)]

(16)

where fi is the activity coefficient of ion i on the mole fraction concentration scale. The experimental mean ionic activity coefficient on molality scale is given by γ± = (γ+ν+ γ−ν− )1/ν , where ν is a stoichiometric coefficient (ν = ν+ + ν− ). These two types of activity coefficients are related as γ± =

f± 1 + 0.018νm

(17)

The solvation entropy is [20] !S =

−3(εW − 1)λ(1 + λ)β6 T(2εW (λ + 1)2 + λ(λ + 3))



 (−Zi2 e2 /2)λ(1 + λ)3 (1 + 3λ) δn (1 − (1/εW )) × + 8ε2W (δi /2 + δn /(2λ)) (2λ2 (δi /2 + δn /(2λ))2 )/2β62

And the Helmholtz energy change !A can be derived as  −2d02 i ρi Zi Ni + 2d0 d2 ρn B10 β !A = − V 12π + J where J=

(18)

(19)

π

2 ρk ρj δ3kj (2l + 1)−1 [hmnl kj (r = δkj )] 3 kj mnl

where hmnl ij is given in [20].From thermodynamic equation, !A = !E − T !S and then  β !A d02 i ρi Zi Ni − 2d0 d2 ρn B10 − 2d22 ρn b2 /δ3n = V 4π − 1/k

 −3(εW − 1)λ(1 + λ)β6 × T(2εW (λ + 1)2 + λ(λ + 3))

 (−Zi2 e2 /2)λ(1 + λ)3 (1 + 3λ) δn (1 − 1/εW ) × + 8ε2W (δi /2 + δn /(2λ)) (2λ2 (δi /2 + δn /(2λ))2 )/2β62 where the internal energy and the entropy were obtained from Eqs. (8) and (18), respectively.

(20)

M. Lotfikian, H. Modarress / Fluid Phase Equilibria 209 (2003) 13–27

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In the PMSA model, according to the Garisto et al. [28], there are two competing terms that influence the solvation energy: !G = !Gin + !Gnm

(21)

the ion–dipole interaction for solvation energy !Gin = −

(Zi2 e2 /2)(1 − 1/εW ) (δi /2) + (δn /(2λ))

(22)

and dipole–dipole solvation energy which is always positive !Gnn =

(Zi2 e2 /2)(1 − 1/εW )2 δn εW (1 + λ)[8δin (1 + λ)/δn − 3λ − 3(1 − 1/λ)/β6 ] 8[λ(λ + 3) + 2εW (1 + λ)2 ][δi /2 + δn /(2λ)]2

(23)

where εW the Wertheim’s dielectric constant for low ionic concentration is given by [19,20] λ2 (1 + λ)4 16 for high ionic concentrations the Adelman’s dielectric constant εA is proposed as [19,20]: εW =

1 + d22 β62 (1 + λ)4 16 The excess Gibbs solvation energy in the PMSA model is as following [36,37]:

(zi e)2 1 ex Gin = − (εW − 1) δi εW 1 + (δn /δiλ ) εA =

(24)

(25)

(26)

and for the dipole term Gex nn



(zi e)2 (εW − 1)2 4 + (δn /δi λ)((3λ + 2)/(λ + 1)) =− 8δi εW (1 + (δn /δi λ))2 (εn + (λ(λ + 3)/2(λ + 1)2 ))

(27)

In electrolyte solutions, the water molecule parameters such as molecular diameter and dipole moment play an important role in the calculating of thermodynamic properties. In the NPSMA model, the water molecule is considered as a hard-sphere dipole and all the other molecular interactions, such as Lennard–Jones, quadruple moment, induced dipole moment and hydrogen bond, are neglected [29]. Various values for the hard-sphere diameter of water have been obtained by fitting the experimental vapour pressure data in the temperature range [24,25,34,35]. In our calculations we used the reported values for diameter and dipole moment by Wei and Blum [20]: δn = 0.27 nm and µ = 2.21 D which are closer to the experimental values for water molecule: 0.295 nm, 1.85 D. 4. Results and discussion The results of calculations are reported in Tables 1–3 and Figs. 1–4. In Table 1, the cation and anion diameters calculated by the NPMSA model for CsCl, as an example are presented. In this table, the reduced chemical potential calculated by NPMSA for cation, anion and solvent are presented. The reduced

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M. Lotfikian, H. Modarress / Fluid Phase Equilibria 209 (2003) 13–27

Table 1 The diameters and chemical potentials of CsCl and the used conditions State

Molality

δ+ (nm)

−βµNPMSA +

−βµNPMSA −

−βµNPMSA n

δPMSA (nm) +

−βµPMSA +

−βµPMSA −

A B C D E F G H

0.01 0.10 0.50 1.00 1.60 2.00 2.50 3.00

1.39208 0.73216 0.50152 0.40220 0.33485 0.30287 0.27089 0.24476

68.6886 82.4245 117.201 138.551 160.612 181.003 200.119 210.877

59.3846 69.2319 99.316 118.512 124.705 138.323 146.645 157.442

49.696 47.437 40.158 35.389 33.130 31.625 30.872 30.621

0.51083 0.35594 0.30422 0.27977 0.24474 0.23569 0.2356 0.22198

70.8866 86.3057 120.457 142.158 165.297 186.059 205.810 216.097

64.3886 74.3431 104.233 123.614 129.539 142.894 150.448 161.321

The subscripts +, − and n denotes the cation, anion and dipolar solvent, respectively. The used conditions are: T = 298.15 K, δn = 0.27, µ = 2.21 D, the total density ρtot δ3 = 0.6585, |Zi | = 1. Table 2 Comparison between the internal and Helmholtz energies for ion–ion terms and the total internal energy of the restricted MSA (RMSA), primitive MSA (PMSA) and non-primitive MSA (NPMSA) State

RMSA

A B C D I F G H

PMSA

NPMSA

−βEcc

−βA

−βEcc

−βA

−βEcc

−βEtot

−βA

0.0239 0.0736 1.2903 4.2896 10.488 14.647 21.587 28.644

0.0343 0.0724 1.2765 4.2502 10.408 14.534 21.430 28.437

0.0258 0.0967 0.9765 2.7664 6.2230 9.7916 11.4125 14.6705

1.4362 1.7122 2.0741 2.8554 4.2481 5.1248 6.5168 7.9715

0.0042 0.0846 0.7614 2.5380 4.8729 6.7680 8.8830 11.167

2.2242 2.9724 3.7959 5.7252 8.1927 10.298 12.517 15.053

1.7615 1.6154 1.3976 1.2549 1.2146 1.1629 1.1534 1.1531

chemical potential calculated by the PMSA for cations and anions are also reported in Table 1. It is worth nothing that the anion diameter in PMSA is constant and is taken as the Pauling diameter (δ+ = 0.362 nm) the cation diameter is an adjustable parameter and is reported in Table 1. As it is seen from the result in Table 1 the calculated ionic diameters are different for the NPMSA and the PMSA. It can be stated that the diameters calculated by the NPMSA model is more accurate than those Table 3 The dielectric constant, entropy, pressure and Gibbs energy from the NPMSA, the PMSA and the MSA State

εA

−!S PMSA /k

−!S NPMSA /k

−βV !P PMSA

−βV !P NPMSA

−β !GPMSA

−β !GNPMSA

A B C D E F G H

72.6181 46.1653 24.9399 16.8134 13.9781 12.3669 11.6345 11.4005

8.0123 10.6455 20.879 25.195 29.699 41.411 48.989 56.957

7.4506 9.5343 18.129 23.796 25.627 35.173 40.433 43.161

2.2430 10.796 13.375 38.762 79.421 111.54 156.57 206.48

0.7611 1.3570 2.3983 4.4703 6.9781 9.1351 11.3636 13.8999

1.3817 7.5071 26.221 42.517 57.260 65.247 73.753 80.978

0.1337 1.7366 2.1973 3.0295 4.4834 5.2912 6.6249 7.9247

M. Lotfikian, H. Modarress / Fluid Phase Equilibria 209 (2003) 13–27

21

Fig. 1. Plot of MSA parameters as a function of ionic concentration for CsCl–H2 O system. T = 298.15 K, δn = 0.27 nm, δ− = 0.362 nm, µ = 2.21 D, |Zi | = 1.

Fig. 2. Plot of the Helmholtz energy change (solid line) and pressure change (dashed line) as a function of dipolar solvent diameter (nm). T = 298.15 K, δn = 0.27 nm, µ = 2.21 D, |Zi | = 1.

Fig. 3. The comparison between cation diameters (Cs in CsCl) in the PMSA (solid line) and the PMSA (dotted line) models vs. molality. T = 298.15 K, δ− = 0.362 nm, δn = 0.27 nm, µ = 2.21 D, |Zi | = 1.

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M. Lotfikian, H. Modarress / Fluid Phase Equilibria 209 (2003) 13–27

Fig. 4. Plot of activity coefficient of NaOH as a function of molality. The experimental activity coefficient (the points), the calculated activity coefficient (the line) T = 298.15 K, δn = 0.27 nm, µ = 2.21 D, |Zi | = 1.

of the PMSA due to neglecting the effect of solvent in the NPMSA. The values of the NPMSA chemical potential are less than those of PMSA, this also can be attributed to the solvent effect, which seems to appear in the solvent parameter Ni in Eq. (11). The chemical potential of the cation and anions terms calculated by the PMSA are larger than those calculated by the NPMSA model. The dipole moment has a dramatic effect on the chemical potentials for high valence ion and dilute solution, and this indirect effect is accounted for in the expression of Ni while the PMSA neglects this effect. So the chemical potential in the PMSA in overestimated [24]. As Table 1 shows the chemical potential in the NPMSA and Table 1 shows the chemical potential in the NPMSA and PMSA models have same trend, that is, on increasing the molality of the electrolyte the ionic chemical potential increases. The reverse is observed on considering the chemical potential of dipole βµNPMSA , which shows a decreasing trend. This behaviour is quite expected by considering the n two terms in Eq. (12) where the first term is positive and the second term is negative. Table 2 shows the Helmholtz energy change βA calculated by Eq. (19), proposed by Wei and Blum [19,20] and by Eq. (20) proposed in this work. The calculated Helmholtz energy by these equations is the same. Also in Table 2, the ion–ion energy change βEcc and the total energy change βEtot are reported. The βEcc and βA where also calculated by the RMSA and the PMSA models for comparison with the NPMSA model. The internal and Helmholtz energies of ion–ion term calculated by the RMSA and the PMSA are larger than those calculated by the NPMSA, these differences as stated before are attributed to the dipole moment effect of solvent which is neglected in the RMSA and the PMSA models. However, comparison of the internal energy and the Helmholtz energy results for the RMSA and the PMSA models show that these values for the RMSA are larger than the PMSA, the reason for this can be explained in terms of inability of the RMSA to consider different ionic diameters for cation and anion. Also Table 2

M. Lotfikian, H. Modarress / Fluid Phase Equilibria 209 (2003) 13–27

23

shows that the NPMSA underestimate the ion–ion energies. This is due to the assumption of definite orientations of dipoles in the vicinity of the ions in the NPMSA model. This point has been shown by computer simulation results by Eggebrecht and Ozler [33]. Table 3 shows the calculated results of the dielectric constant, the entropy, the pressure and the Gibbs energy for the NPMSA and the PMSA models. The PMSA results were calculated by Eqs. (26) and (27). Our calculations indicate that the Wertheim’s dielectric constant; εW in Eq. (24) is equal to the Adelman’s dielectric constant εA in Eq. (25). The results in Table 3 show that higher entropy values are calculated on increasing the concentration of electrolyte. The reason for this can be explained in terms of higher disorder in the solution due to the interaction of ion with the water molecules. These interactions destroy the structural order of water molecules due to the hydrogen bondings. It is obvious that these effects are not properly taken into account in the PMSA model, therefore the calculations lead to higher entropy, pressure and Gibbs energy values for the electrolyte system, compared with the NPMSA model. By fitting the cation diameters as adjustable parameters to the experimental data for single ionic activity a simple equation has been derived for calculating the ionic sizes up to 6 M for aqueous electrolyte solutions. This equation is δi = −a ln(I) + b

(28)

In this equation the cation coefficients a and b are defined as b = a ln(I0 ) + C

(29)

where I0 is the ionic strength at infinite dilution and I is defined as I (mol/kg−1 ) =

1
mi Zi2 2

(30)

At infinite dilution I = I0 and δi = C, where C is the cationic diameter at infinite dilution. In Table 4, the parameters a and b in Eq. (29) for cation diameter obtained by correlating with the experimental mean ionic activity coefficients of the NPMS model, Eqs. (15) and (16) are reported for anion due to weak hydration [32] the effect of concentration variations on the anion diameter can be neglected. The values of the parameters show that the cationic diameter varies from one salt to the other. These variations are expected and have been explained in terms of the effects of adjacent anions [25,32]. Also these variations can have a dramatic effect on calculating the thermodynamic properties of mixed electrolytes. Also in Table 4 the average relative deviation percent (ARD %) in the calculation of activity coefficients by the NPMSA, the PMSA and the Pitzer model are reported. The anion diameter in the NPMSA and the PMSA models is assumed to be independent of concentration but the cation diameter is expressed in terms of three adjustable parameter [31,32]. In fact, in these models and also Pitzer model the lone ion diameters are not involved and the hydrated ion diameter are considered. However, the hydration of anion in these models is supposed to be weak and can be neglected, but the hydration of cation is strong and the hydration layer must be included in the calculations. Fig. 1 shows the variation of parameters in Eq. (12) to justify the dipole chemical potential variations. The overall explanation is that on increasing the concentration of electrolyte the ion–dipole interactions cause a decrease in the chemical potential of dipole.

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M. Lotfikian, H. Modarress / Fluid Phase Equilibria 209 (2003) 13–27

Table 4 The diameter constants (a, b) of the cations and the relative deviations in γ± and compared with PMSA and Pitzer results System (salt + H2 O)

HCl NaCl KCl CsCl LiCl NH4 Cl NaBr KI KNO3 AgNO3 NaOH SrCl2 CaCl2 MgCl2 K2 SO4 H2 SO4 Na2 S2 O3 BaCl2 AlCl3 ZnSO4

Cationic parameters (nm)

ARD %

mmax

a

b

PMSA

Pitzer

Present work

0.1122 0.1433 0.1346 0.1433 0.1777 0.1301 0.1573 0.1339 0.1470 0.1471 0.1692 0.1670 0.1866 0.2522 0.1862 0.1483 0.1633 0.1766 0.2669 0.2024

0.2689 0.3634 0.3929 0.4022 0.2838 0.4122 0.3446 0.3917 0.4287 0.4350 0.3464 0.5719 0.5677 0.7437 0.6972 0.6546 0.6675 0.7040 0.9821 0.9628

28.34 2.11 2.00 0.33 28.34 3.09 3.14 2.83 2.60 – 1.72 4.29 22.33 18.52 0.23 – – 1.32 6.00 7.47

10.08 0.68 0.56 0.75 10.08 0.90 3.14 0.22 0.46 – 1.50 1.22 4.59 1.83 0.53 – – 0.26 2.49 5.04

13.479 1.919 1.621 0.522 11.211 1.766 0.537 0.489 0.325 0.215 0.884 3.002 6.951 3.112 0.965 0.333 0.244 0.541 0.669 0.798

19.1 6.1 5.0 5.0 19.2 7.4 9.0 4.5 3.5 6.0 10.0 4.0 7.5 5.9 0.7 6.0 3.5 1.8 1.8 3.5

In the PMSA the anion diameter is constant (Pauling diameter) and the constants of cationic diameter are those reported in [32].

It is interesting to know how the Helmholtz energy changes with the dipole diameter. As we can see in Fig. 2, on increasing the dipole diameter the Helmholtz energy goes to the positive values and then from about 3 nm becomes constant. This can be related to the domination of repulsive interactions in the higher dipole diameter. Also Fig. 2 shows that the larger dipole diameter the interactions in electrolyte solution cause the pressure change (!P) tends towards a limit. Fig. 3 shows the variation of hydration diameter of cation in the NPMSA and the PMSA models. In higher concentration the cation diameter shows a decrease due to the thinner hydration layer. As has been observed by the other researchers [24,25,32], Fig. 3 shows that the cationic diameter in the NPMSA depends strongly on the ionic strength. Fig. 4 shows the experimental mean ionic activity coefficients versus molalities. This figure shows that there is a good agreement between the calculated and experimental mean ionic activity coefficients. To convert molar concentration of electrolyte to molalities, in our calculations, the density of electrolyte solution has been obtained from [30]. The minimum ARD % of the mean ionic activity coefficient, which is expressed in the following form, is taken as the objective function in the calculations. N

100
|γcal (k) − γexp (k)| ARD % = N k=1 γexp (k)

(31)

M. Lotfikian, H. Modarress / Fluid Phase Equilibria 209 (2003) 13–27

25

where N is the number of concentration points and γexp (k) is the experimental mean ionic activity coefficient.

5. Conclusion The Helmholtz and internal energies of electrolyte solution were calculated by the RMSA, the PMSA and the NPMSA models. The other thermodynamics properties such as entropy and Gibbs energy were calculated by the NPMSA model and compared with the results of calculations by the PMSA model. The differences between the calculated properties by various models were discussed and it was concluded due to neglecting the dipole moment effects in the other models (PMSA, RMSA) the PMSA model calculates the correct properties which is lower than those calculated by the other models. The cationic diameter was expressed in terms of a two-constant function of electrolyte concentration. The constants of this function were obtained by correlating with the experimental data for the mean activity coefficient of electrolyte solutions. The results indicated that the PMSA model calculated the activity coefficient with good accuracy. The osmotic coefficient is an important property for electrolyte solutions and can be used to evaluate the predictive ability of the proposed electrolyte models. In the future work, it is intended to propose a feasible approach for calculating the osmotic pressure in the NPMSA model. List of symbols A Helmholtz energy b2 , B10 MSA parameters C direct pair correlation function D density of solution (kg m−3 ) e elementary charge (C) E internal energy f ionic activity coefficient in mole scale g radial distribution function G solvation Gibbs energy h total pair correlation function H enthalpy I ionic strength I0 infinite dilution ionic strength k Boltzmann’s constant m molality concentration (mol kg−1 ) P pressure (Pa) r radial distance (nm) S entropy T absolute temperature (K) U pair potential V volume (m3 ) x mole fraction Z ionic charge

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Greek letters ␤ 1/kT δ particle diameter (nm) ε dielectric constant rotational invariant Φmnl γ ionic activity coefficient in molality scale Γ MSA parameter ν stoichiometric coefficient ρ number density Superscripts ex excess hs hard-sphere MSA mean spherical approximation Subscripts A i, j in n nn w W − + ±

Adelman component indices interaction between ion and dipole dipole interaction between dipole and dipole water Wertheim anion cation mean ionic

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