Modeling Sorbent Phase Nonideality for the Accurate Prediction of

Article pubs.acs.org/jced

Modeling Sorbent Phase Nonideality for the Accurate Prediction of Multicomponent Ion Exchange Equilibrium with the Homogeneous Mass Action Law José P. S. Aniceto, Patrícia F. Lito, and Carlos M. Silva* Department of Chemistry, CICECO, University of Aveiro, Campus Universitário de Santiago, 3810-193, Aveiro, Portugal ABSTRACT: In this Article the ion exchange equilibrium of binary and multicomponent systems was modeled with homogeneous mass action law using activities and taking account of the partial dissociation of salts according to Kester and Pytkowicz's approach. The activity coefficients in the solution were estimated by the Pitzer model and in the sorbent by Wilson, nonrandom two-liquid (NRTL), and universal quasichemical (UNIQUAC) models. Wilson has already been adopted in the literature, while NRTL and UNIQUAC were investigated here for the first time. The modeling procedure relied on the analysis of experimental data for the constitutive binary systems to fit the equilibrium constants and the binary parameters of the activity coefficients for the sorbent, which were then applied to predict the multicomponent isotherms. The models were tested with 22 binary, 14 ternary, 5 quaternary, and 1 quinary systems, totalizing 1494 points. The results showed that NRTL and Wilson provide reliable and very similar predictions (average deviations equal to 11.35 % and 11.53 %, respectively) while UNIQUAC performs poorly (16.02 %). The ideal model, which assumes unitary activity coefficients, achieved the largest error (34.20 %), thus emphasizing the chief importance of nonidealities. The extension of ion association also proved to be very important, attaining values near 60 %.

■

concentration and electric potential gradients,7−17 the last one being particularly advantageous at high concentrations.8,17 Concerning ion exchange equilibrium, the models adopted for its representation and/or prediction may be divided into four groups:8 (i) homogeneous mass action models, which describe equilibrium as stoichiometric processes by using mass action law, assuming the exchanger as homogeneous and introducing the nonidealities via activity coefficients of ions in solution and in the solid phases; (ii) heterogeneous adsorption models, which treat ion exchange as an adsorption process and explain deviations from ideal behavior in terms of energetic heterogeneity of the functional groups of the ion exchanger; (iii) heterogeneous mass action models, derived from the original work of Melis et al.,18 are also based on the mass action law but takes the heterogeneity of ion exchanger sites into account; and (iv) purely empirical models, which are used with correlative purposes only. Several researchers have adopted the mass action law approach, for example, Dranoff and Lapidus,20 Pieroni and Dranoff,21 Klein et al.,22 Smith and Woodburn,23 Sengupta and Paul,24 Shallcross et al.,25 Martinez et al.,26 Mehablia et al.,27 Ioannidis et al.,28 Vo and Shallcross,29 Mumford et al.,30 Borba et al.,31 and Aniceto et al.19 The deviations from ideality in electrolyte solutions are important even at low concentrations. Several models may be found in the literature to estimate them, the Debye−Hückel

INTRODUCTION Ion exchange is one of the most effective and thus commonly applied techniques for the removal of ionic contaminants from waters and wastewaters as well as for the recovery of valuable metals.1−3 It may be also used to preconcentrate metals whenever their concentrations are so low that a previous preconcentration step is mandatory, as for instance in metal analysis.2 The modeling and simulation of an ion exchange process is fundamental to the study and design of several applications, including the softening and deionization of water, wastewater treatment, catalysis, chemical purification, plating, and food and pharmaceutical uses. The prediction of their dynamic behavior and optimization of their operating conditions, as well as scaling up to industrial level, may be advantageously accomplished by computer simulation to reduce associated costs and the number of indispensable experiments. At this point it should be stressed that the real usefulness of a model is mainly related with its ability to predict the process behavior under operating conditions different from those used to obtain its parameters. Kinetic and equilibrium data are both necessary for successful simulations. In the literature, ion exchange kinetics is frequently interpreted by semiempirical pseudo-first and pseudosecond order equations,4−6 which possess no strict theoretical background and, consequently, have restricted application and extrapolation capability. Instead, both Nernst−Planck and Maxwell−Stefan models are able to effectively describe the mass transport in ionic systems, since they account for both © 2012 American Chemical Society

Received: February 6, 2012 Accepted: April 18, 2012 Published: May 15, 2012 1766

dx.doi.org/10.1021/je300156h | J. Chem. Eng. Data 2012, 57, 1766−1778

Journal of Chemical & Engineering Data

Article

and Pitzer models being the most frequently used.16,23,25,31−33 Notwithstanding, de Lucas et al.34 and Martinez et al.26 selected the Meissner and Kusik electrolyte solution. Besides, Vo and Shallcross35 investigated the binary and multicomponent equilibria of H+, Na+, K+, Mg2+, and Ca2+ ions in Amberjet 1200H, using the Wilson model for the resin, and comparing the performance of three models for solution activity coefficients. The corresponding order of increasing accuracy was found to be: ideal < extended Debye−Hückel < Meissner and Kusik < Pitzer.35 Concerning the solid phase, the expressions of Wilson and Margules8,36,37 have been mostly applied to compute the activity coefficients (e.g., Smith and Woodburn,23 Vamos and Haas,38 Shehata et al.,39 de Lucas et al.,33 Aniceto et al.19). The number of research works in the literature concerning multicomponent equilibria is quite scarce in comparison to those on binary systems, given the large complexity arisen from the diverse background processes causing nonidealities and coupling of their models. For instance, the nature of the systems, the interactions of counterions in the solution and in the solid phase, the interactions between them and the exchanger, the heterogeneity of the exchanger surface, clustering and dissociation of exchangeable ions, limited solubility of counterions, and the synergistic effect of competitive ions are some of those complexities.40 The general approach adopted by researchers for multicomponent analysis concerns the development of good semitheoretical models based on binary ion exchange data. Those models are first applied to binary equilibrium data to optimize the corresponding binary parameters, which are then used in the extended formulation to predict multicomponent equilibria. According to Shallcross41 any ion exchange equilibrium model should satisfy the following requirements: (i) model parameters must be independent of solution phase concentration; (ii) model parameters must remain constant in the presence of different nonexchanging counterions; (iii) internal consistency of the model is essential, that is, models should be not overspecified and the parameters values must not be significantly dependent on which equations are used; and (iv) the model must be applicable up to quaternary systems. Smith and Woodburn23 and Shehata et al.39 used the Wilson model for the solid phase and the extended Debye−Hückel equation for the solution to study the equilibrium behavior of SO42−/NO3−/Cl− in Amberlite IR-400 and Sr2+/Cs+/Na+ on chabazite, respectively. Shallcross et al.25 proposed an identical model for Ca2+/Mg2+/Na+ equilibrium on Amberlite 252 but estimated the aqueous phase activity coefficients by Pitzer theory inasmuch as the extended Debye−Hückel equation did not take the nature of other coexisting ions into account, except through their effect upon the ionic strength. de Lucas et al.33 studied the Na+/K+/Amberlite IR-120 equilibrium on different organic and mixed solvents, with the Debye−Hückel and Wilson models to represent activity coefficients in solution and solid phases, respectively, while Vamos and Haas38 adopted the Margules equations to compute the resin phase activity coefficients. The statistical analysis of data fits indicated that even though both Wilson and Margules equations were able to model adequately the resin phase nonidealities, the Wilson model provided better correlations. Recently, the Bromley and the Wilson models were selected by Borba et al.31 for solution and solid nonidealities, respectively, being able to predict the ternary system Cu2+/Zn2+/Na+ in Amberlite IR-120.

Besides the nonideality effects, the real concentrations of all ionic species present in solution must be appropriately considered to correctly describe equilibrium. Once the energy of the mutual electrical attraction between close ions of opposite sign may be considerably greater than their thermal energy, new stable entities may be formed in solution.8,42,43 As a result, ion association phenomenon should be taken into account, particularly at high concentrations, since the number of free ions available for ion exchange is reduced. This approach is recurrently discarded because of the lack of the equilibrium constants necessary to represent the association equilibria. The parameters involved in calculations for a binary system, that is, the equilibrium constant and the activity coefficient model parameters for the exchanger phase, frequently exhibit a disturbing interdependence during their simultaneous correlation.23,25 Accordingly, alternative approaches to decouple their values to get equilibrium constants independent of the solution concentration have been adopted. For instance, the approaches of Argersinger et al.,8,44 Gaines and Thomas,8,45 and Ioannidis et al.8,28 have been successfully applied with that purpose. An important example is the model proposed by Mehablia and coworkers for multicomponent systems,27,46 which computes the nonidealities of solution and exchanger phases by applying Pitzer's electrolyte solution theory and the Wilson model, respectively, and adopts the equilibrium-based approach of Kester and Pytkowicz43 for the ionic pairing effect. By employing the Gaines and Thomas45 approach, they decoupled the calculation of the binary equilibrium constants from that of the Wilson parameters. In this work the ion exchange equilibrium in multicomponent systems is studied under the scope of the mass action law to evaluate, for the first time, the applicability of the nonrandom two-liquid (NRTL) and universal quasichemical (UNIQUAC) models to quantify nonidealities in the exchanger phase. Up until now, only the Wilson equation has been adopted to calculate the activity coefficients in the solid; hence it is important to evaluate the ability of other models to embody the representation of binary systems for the subsequent prediction of multicomponent ion exchange equilibrium. The NRTL and UNIQUAC models have been selected in this essay since they are based on binary parameters only, which means multicomponent mixtures can be estimated using those pair constants. NRTL and UNIQUAC models have been recently applied but only to binary systems, providing reliable results.19 The validation of our global multicomponent model is accomplished here with experimental data taken from the literature for very distinct systems, in terms of counterion valences and total salt concentrations. The outline of the paper is as follows. First, the fundamentals of mass action law, the models for activity coefficients of ions in solution and exchanger, the Kester and Pytkowicz's approach for ion association, and the Ioannidis et al. method to determine noncorrelated parameters of the global model are described. Then the database containing the experimental points collected from the literature is listed. In the following, the binary correlation and multicomponent prediction results obtained with mass action law combined with Pitzer and NRTL/UNIQUAC/Wilson models are presented and discussed. Finally, the most important conclusions of the work are drawn. 1767



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2. MODELING ION EXCHANGE EQUILIBRIUM 2.1. Mass Action Law. The ion exchange equilibrium of counterions AzA and BzB, with valences zA and zB, may be represented by: z B AzA + zA B z B ↔ z B AzA + zA B z B

model of Pitzer was adopted, since it provides very good results up to 6 molal.8,47 The Pitzer equations for a particular cation, γc , and a particular anion, γa , are:47,48 n

ln γc = zc 2F +

(1)

j=1 j≠c

where the top bar identifies the exchanger phase, and the modulus was omitted in the stoichiometric coefficients for simplicity. The equilibrium constant is defined in terms of the counterion activities ai and ai̅ by: KBA(T ) =

a A̅ z BaBzA aAz BaB̅ zA

+

+

=

z B A + zA B

zB

↔ z B A + zA B

zB

(6)

zC AzA + zA CzC ↔ zC AzA + zA CzC

(7)

a A̅ z BaBzA aAz BaB̅ zA

KBC =

aC̅ z BaBzC aCz BaB̅ zC

+

+

∑ j=1 j≠i

∑ mj(2Θaj + ∑ mi Ψaji) i=1 i≠a n

∑ ∑ mimj(za 2Bij′ + |za|Cij)

+

1 2

∑ ∑ mimi′Ψii′ a i = 1 i ′= 1 i ≠ a i ′≠ a

(11)

where:

I=

a A̅ zCaCzA aAzCaC̅ zA

Only two equations are independent, since the three equilibrium constants satisfy the so-called triangle rule, (KBC)zA (KAC)−zB (KAB)zC = 1. Once more, these results may be extended to any ion exchange system involving nc > 3 counterions, Azi i. In a multicomponent ideal system, when the selectivity coefficients are known, the mole fraction of any counterion yi in exchanger may be expressed in terms of xi by: n

j=1 j≠a

i=1 j=1 i≠a j≠a n n

(8)

yi +

n

∑ mi{2Bia + (2 ∑ mjzj)Cia} n

j=1 j≠a n

F=−

K CA =

(10)

n

whose equilibrium constants are: KBA =

j = 1 j ′= 1 j ≠ c j ′≠ c

i=1 i≠a

(5)

z BCzC + zC B z B ↔ z B CzC + zC B z B

∑ ∑ mjmj ′Ψcjj ′ n

(4)

zA

1 2

(3)

yAz B mBzA mAz ByBzA

n

ln γa = za 2F +

In the case of multicomponent systems, the previous equations are easily extended to nc exchangeable counterions. For instance, for the ternary system AzA/BzB/CzC the following equilibria are now implied: zA

j=1 j≠c

∑ ∑ mimj(zc 2Bij′ + |zc|Cij)

+

while for ideal systems (γi̅ = γi = 1) the well-known selectivity coefficient is obtained: Kc(T , mi , yi ) =

∑ mi(2Θci + ∑ mj Ψcij)

i=1 j=1 i≠c j≠c n n

The activity is the product of concentration by the activity coefficient on the same concentration scale. Since mole fractions, yi, are usually adopted for the ion exchanger, and molalities, mi, for the solution, one writes: ai̅ = γi̅ ·yi and ai = γi·mi. In the case of ideal exchanger (γi̅ = 1), eq 2 reduces to the corrected selectivity coefficient, KAaB: A K aB (T , yi ) = KBA ×

n

i=1 i≠c

(2)

yAz B aBzA aAz ByBzA

i=1 i≠c

n

n

⎛ γ ̅ zA ⎞ ⎜⎜ Bz ⎟⎟ ⎝ γ A̅ B ⎠

n

∑ mj{2Bcj + (2 ∑ mizi)Ccj}

1 2

⎛ yi ⎞ zj / zi =1 ⎜ ⎟ z /z −1 Kc,1/i /zjimt j i ⎝ xi ⎠

n

∑ zi 2mi

(13)

i=1

(14)

Bij = βij(0) + βij(1)g (α1 I ) + βij(2)g (α2 I )

(15)

Cij = (9)

(12)

⎛ e 2 ⎞3/2 2πρw N0 Aγ = ⎜ ⎟ 1000 ⎝ εkT ⎠

B′ij =

xj

Aγ ⎡ ln(1 + 1.2 I ) ⎤ I + ⎢ ⎥ ⎦ 3 ⎣ 1 + 1.2 I 0.6

βij(1)g ′(α1 I ) + βij(2)g ′(α2 I ) I

(16)

Cij(0) 2 |zizj|0.5

(17)

Here the subscripts i and j symbolize all cations and anions in solution, respectively, and c and a identify the specific cation and anion under calculation. The ionic strength, defined by eq 13, is calculated by summing the effect of all n ionic species in solution. In the Aγ term, N0 is the Avogadro's constant, ρw and ε are density (g·cm−3) and dielectric constant (C2·N−1·m−2) of pure solvent, k = 1.38066·10−23 J·K−1 is Boltzmann's constant, and

where xi is the mole fraction of counterion Azi in solution and mt the total molality of ionic species. 2.2. Solution Phase Modeling. Several expressions may be used to estimate activity coefficients of ionic species in solution, namely, Debye−Hückel, Hückel, Bromley, Guggenheim, Pitzer, Meissner and Kusik, and so forth.8,36,47 In this work, the milestone 1768



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Table 1. Database of Experimental Ion Exchange Equilibria at 298.15 K system Binary Systems Na+/H+/Cl−/Amberjet 1200H29 K+/Na+/Cl−/Amberjet 1200H29 K+/H+/Cl−/Amberjet 1200H29 Ca2+/H+/Cl−/Amberjet 1200H29 Ca2+/Na+/Cl−/Amberjet 1200H29 Ca2+/K+/Cl−/Amberjet 1200H29 Ca2+/Mg2+/Cl−/Amberjet 1200H29 Mg2+/H+/Cl−/Amberjet 1200H29 Mg2+/Na+/Cl−/Amberjet 1200H29 Mg2+/K+/Cl−/Amberjet 1200H29 Ca2+/Na+/Cl−/Clinoptilolite57 Ca2+/K+/Cl−/Clinoptilolite57 K+/Na+/Cl−/Clinoptilolite57 Ca2+/Na+/Cl−/Amberlite 25225 Mg2+/Na+/Cl−/Amberlite 25225 Ca2+/Mg2+/Cl−/Amberlite 25225 Cs+/Na+/Cl−/Dowex 50W-X858 Mn2+/Na+/Cl−/Dowex 50W-X858 Mn2+/Cs+/Cl−/Dowex 50W-X858 Zn2+/H+/NO3−/Amberlite IR 12024 Cd2+/H+/NO3−/Amberlite IR 12024 Cd2+/Zn2+/NO3−/Amberlite IR 12024 Ternary Systems K+/Na+/H+/Cl−/Amberjet 1200H29 K+/Na+/Ca2+/Cl−/Amberjet 1200H29 K+/Na+/Mg2+/Cl−/Amberjet 1200H29 Na+/H+/Ca2+/Cl−/Amberjet 1200H29 Na+/H+/Mg2+/Cl−/Amberjet 1200H29 Na+/Mg2+/Ca2+/Cl−/Amberjet 1200H29 K+/H+/Ca2+/Cl−/Amberjet 1200H29 K+/H+/Mg2+/Cl−/Amberjet 1200H29 K+/Mg2+/Ca2+/Cl−/Amberjet 1200H29 H+/Mg2+/Ca2+/Cl−/Amberjet 1200H29 K+/Na+/Ca2+/Cl−/Clinoptilolite59 Ca2+/Mg2+/Na+/Cl−/Amberlite 25225 Na+/Cs+/Mn2+/Cl−/Dowex 50W-X858 Cd2+/Zn2+/H+/NO3−/Amberlite IR 12024 Quaternary Systems K+/Na+/Ca2+/Mg2+/Cl−/Amberjet 1200H29 K+/Na+/H+/Ca2+/Cl−/Amberjet 1200H29 K+/Na+/H+/Mg2+/Cl−/Amberjet 1200H29 K+/H+/Ca2+/Mg2+/Cl−/Amberjet 1200H29 Na+/H+/Ca2+/Mg2+/Cl−/Amberjet 1200H29 Quinary System K+/Na+/H+/Ca2+/Mg2+/Cl−/Amberjet 1200H29

g ′(x) =

Ct/N

NDP 19 18 18 19 17 15 18 15 14 15 19 19 14 7 8 7 13 15 13 16 15 13

0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.05 0.05 0.05 0.05 0.05 0.05 0.1 0.1 0.1 0.1 0.1 0.1

76 68 74 68 68 67 69 68 65 65 29 32 15 15

0.1, 0.2, 0.1, 0.2, 0.1, 0.2, 0.1, 0.2, 0.1, 0.2, 0.1, 0.2, 0.1, 0.2, 0.1, 0.2, 0.1, 0.2, 0.1, 0.2, 0.05 0.1, 0.2 0.1 0.1

0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5,

1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0

60 63 60 62 66

0.1, 0.1, 0.1, 0.1, 0.1,

0.5, 0.5, 0.5, 0.5, 0.5,

1.0 1.0 1.0 1.0 1.0

77

0.1, 0.2, 0.5, 1.0

nc j=1

yk Λki nc ∑ j = 1 yj Λkj k=1

∑

(20)

⎛ λij − λii ⎞ exp⎜ − ⎟ vi RT ⎠ ⎝

vj

Λij =

0.2, 0.2, 0.2, 0.2, 0.2,

(19)

nc

ln γi ̅ = 1 − ln(∑ yj Λij) −

(21)

where Λij and Λji are binary interaction temperature-dependent parameters defined such that i ≠ j, being calculated as a function of the pure-component molar volumes (vi,vj) and characteristic energy difference (λij − λii), and R is the ideal gas constant. NRTL Model. The NRTL model is also expressed in terms of binary parameters as follows: n

ln γi ̅ =

∑ j =c 1 yj τjiGji n

∑l =c 1 yl Gli

nc

+

∑ j=1

n ⎛ ∑r =c 1 yr τrjGrj ⎞ ⎜ ⎟ − τ ij n n ∑l =c 1 yl Glj ⎟⎠ ∑l =c 1 yl Glj ⎜⎝

yj Gij

(22)

τij =

gij − gjj

Gij = exp( −αijτij)

RT

(αji = αij) (23)

where gij an energy parameter characteristic of the i−j interaction, and αij is related to the nonrandomness in the mixture (when αij = 0, the binary mixture is completely random); αij varies frequently between 0.20 and 0.47 and can be set arbitrarily when data are scarce (e.g., αij = 0.3).36 UNIQUAC Model. The UNIQUAC model results from a linear combination of a combinatorial term (ln γCi̅ ) accounting essentially for molecule size and shape differences, and a residual term (ln γRi̅ ) resultant from short-range molecular energetic interactions:38,50,51 ln γi ̅ = ln γi̅ C + ln γi̅ R ln γi̅ C = ln

ϕi yi

+

(24)

ϕ θ z qi ln i + li − i yi 2 ϕi

⎡ nc ln γi̅ R = qi⎢1 − ln(∑ θτ j ji) − ⎢⎣ j=1

θi =

qy

ii n ∑ j =c 1 (qjyj )

τij = exp( −aij /T )

2[1 − (1 + x)exp(−x)] x2

x2

2.3. Exchanger Phase Modeling. The applicability of NRTL and UNIQUAC models to represent the activity coefficients of ions in the exchanger phase for the case of multicomponent systems will be analyzed in this work for the first time. Their ability to predict multicomponent equilibrium will be compared with that achieved by the Wilson equation, since it has been the unique expression adopted up until now with this purpose. In the following, the three models are summarily presented.8,36,49 Wilson Model. The Wilson equation encloses the effect of both differences in molecular sizes and intermolecular forces, being dependent on binary parameters only. For a solution of nc exchanging species is given by:

e = 1.60206·10−19 C is the electron charge. The second virial coefficients, B, represent the specific binary, near-field interactions between pairs ij, while the corresponding ternary interactions ijk are described by the third virial coefficient, Ψ. Parameters α1 and α2 are dependent on the electrolyte nature; for salts containing a monovalent ion, α1 = 2 and α2 = 0, while for higher valence types the corresponding values are α1 = 1.4 and α2 = 12. The functions g(x) and g′(x), where x ≡ α(I)1/2 are given by: g (x ) =

−2[1 − (1 + x + 0.5x 2)exp(−x)]

(18)

ϕi =

nc

∑ j=1

nc

∑ yj lj j=i

⎤ ⎥ n ∑kc= 1 θkτkj ⎥⎦

(25)

θτ j ij

(26)

ry ii nc ∑ j = 1 (rjyj )

li =

z (ri − qi) − (ri − 1) 2 (27)

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Table 2. Calculated Results for the Binary Ion Exchange Isotherms of Database (Table 1) range of ionic strength Ct/N

range of Y1a

Na+/H+/Cl−/Amberjet 1200H

model for γi̅

0.1

0.10−0.92

K+/Na+/Cl−/Amberjet 1200H

0.1

0.03−0.97

K+/H+/Cl−/Amberjet 1200H

0.1

0.14−0.98

Ca2+/H+/Cl−/Amberjet 1200H

0.1

0.69−0.97

Ca2+/Na+/Cl−/Amberjet 1200H

0.1

0.72−0.99

Ca2+/K+/Cl−/Amberjet 1200H

0.1

0.45−0.99

Ca2+/Mg2+/Cl−/Amberjet 1200H

0.1

0.10−0.89

Mg2+/H+/Cl−/Amberjet 1200H

0.1

0.73−1.00

Mg2+/Na+/Cl−/Amberjet 1200H

0.1

0.80−0.99

Mg2+/K+/Cl−/Amberjet 1200H

0.1

0.64−0.99

Ca2+/Na+/Cl−/Clinoptilolite

0.05

0.10−0.82

Ca2+/K+/Cl−/Clinoptilolite

0.05

0.00−0.66

K+/Na+/Cl−/Clinoptilolite

0.05

0.23−0.97

Ca2+/Na+/Cl−/Amberlite 252

0.05

0.49−0.98

Mg2+/Na+/Cl−/Amberlite 252

0.05

0.60−0.95

Ca2+/Mg2+/Cl−/Amberlite 252

0.05

0.48−0.82

Cs+/Na+/Cl−/Dowex 50W-X8

0.1

0.66−0.97

Mn2+/Na+/Cl−/Dowex 50W-X8

0.1

0.38−0.96

Mn2+/Cs+/Cl−/Dowex 50W-X8

0.1

0.47−0.89

Zn2+/H+/NO3−/Amberlite IR 120

0.1

0.13−0.98

Wilson NRTL UNIQUAC Wilson NRTL UNIQUAC Wilson NRTL UNIQUAC Wilson NRTL UNIQUAC Wilson NRTL UNIQUAC Wilson NRTL UNIQUAC Wilson NRTL UNIQUAC Wilson NRTL UNIQUAC Wilson NRTL UNIQUAC Wilson NRTL UNIQUAC Wilson NRTL UNIQUAC Wilson NRTL UNIQUAC Wilson NRTL UNIQUAC Wilson NRTL UNIQUAC Wilson NRTL UNIQUAC Wilson NRTL UNIQUAC Wilson NRTL UNIQUAC Wilson NRTL UNIQUAC Wilson NRTL UNIQUAC Wilson

system

mol·kg−1 0.1

0.1

0.1

0.15−0.1

0.15−0.1

0.15−0.1

0.15

0.15−0.1

0.15−0.1

0.15−0.1

0.075−0.05

0.075−0.05

0.05

0.075−0.05

0.075−0.05

0.075

0.1

0.15−0.1

0.15−0.1

0.15−0.1 1770

AARD (%) KAB 2.9531 2.9396 14.4329 1.2032 1.2064 1.2115 3.8969 3.8928 42.7306 76.9877 80.7179 3452.6 29.7237 43.4198 68.7108 10.7833 10.7736 10.3821 1.7873 1.8915 1.8267 85.0477 84.5778 629.0 75.6379 57.2310 208.9 22.7630 22.2740 26.6581 0.2108 0.1932 0.2482 2.289·10−4 2.051·10−4 1.974·10−4 32.5048 32.3305 32.8270 4.2683 4.4679 4.0199 2.6842 2.9671 11.5818 2.7550 3.0567 2.3193 23.5328 26.3184 34.8124 1.1270 1.0538 1.8022 6.4513 6.2843 5.1070 18.4918

parameters of γi̅ a 3.5324 5265.4 25641 2.4473 2142.4 −296.33 1.0721 −1659 41116 2.3227 2327.1 1193 4.1736 4381.4 −679.44 0.8025 −1796.2 185.19 4.3114 3575.7 903.34 1.3894 13.7 923.8 0.3196 4364.6 6043.6 2.0187 1539.4 9903.0 0.6243 −5429.8 −899.54 0.1766 −5848.2 −759.68 1.6698 −1434.2 −81.92 2.5308 6406.1 5965.33 2.6687 6576.2 5963.0 5.9116 3961.0 1338.528 3.4331 6372.7 −519.32 0.3561 −2552.7 −888.83 2.1047 −709.4 5340.09 2.5369

0.069 −3863.4 9065 0.4086 −2459.9 880.24 1.6798 206.3 7839 0.4305 −2585.7 4162.8 0.3537 −5188.8 −574.13 1.4667 1628.7 −1.15 0.3433 −4526.9 −903.34 1.1170 −1092.0 4538.1 3.1291 −3406.5 5753.9 0.9429 −2776.8 18445.0 4.8695 3915.6 −333.13 5.6619 4732.9 759.68 1.7940 −1385.3 −406.17 0.1010 −3384.1 −213.40 0.0503 −3615.7 6067.4 0.1692 −4910.9 −1338.53 0.1155 −4428.6 4194.54 2.8079 2150.9 888.83 1.6634 −2379.1 132.14 0.0478

new modela 1.80 1.53 18.56 1.28 1.54 1.84 0.13 0.12 13.82 0.43 0.45 0.95 0.09 0.50 0.12 0.13 0.13 0.25 0.11 1.61 2.03 0.08 0.08 0.33 0.02 0.02 3.30 0.04 0.05 3.80 6.72 5.68 6.73 13.30 16.32 15.01 1.84 1.83 1.88 1.01 0.87 1.71 0.77 0.97 12.94 2.14 2.42 2.25 1.90 1.87 1.92 8.21 7.91 7.92 3.08 3.09 3.23 16.68

ideal 8.18

5.43

9.86

1.99

6.52

2.17

111.92

1.17

0.86

4.94

32.13

92.61

15.23

2.17

1.74

26.98

2.23

7.18

8.10

44.80



Article

Table 2. continued range of ionic strength Ct/N

system

range of Y1a

Cd2+/H+/NO3−/Amberlite IR 120

0.1

0.42−0.93

Cd2+/Zn2+/NO3−/ Amberlite IR 120

0.1

0.18−0.92

model for γi̅ NRTL UNIQUAC Wilson NRTL UNIQUAC Wilson NRTL UNIQUAC

AARD (%)

mol·kg−1

0.15−0.1

0.15

parameters of γi̅ a

KAB 18.7707 344.83 47.9693 45.1325 214.8 1.0346 1.0557 1.0598

7131.1 6790.7 3.1711 810.4 742.5499 0.1335 −2431.1 332.16

−3309.0 5911.4 1.5527 −4218.8 −1270.68 2.0459 4704.3 5453.28

new modela 16.42 48.52 1.93 1.92 2.14 4.44 4.26 4.37

ideal

12.74

49.39

Y1, equivalent fraction of the first counterion of the system. Parameters 1 and 2 of γi̅ : Λ12 and Λ21 (Wilson), Δg12 = g12 − g22 and Δg21 = g21 − g11 (NRTL), a12 and a21 (UNIQUAC). Model: mass action law, ion association, γi by Pitzer, and γi̅ computed by Wilson/NRTL/UNIQUAC equations. a

Table 3. Grand Average Absolute Relative Deviations (AARDs) Obtained with Nonideal and Ideal Mass Action Law system

no. systems

NDP

Wilson

binaries multicomponent total

22 20 42

327 1167 1494

3.21 11.53 10.66

Here z is the coordination number (generally assumed as 10), ϕi and θi are the molecular volume and the surface fractions of component i, respectively, ri and qi are the van der Waals volume and the external surface area parameters, τji is a temperature-dependent parameter, and aij is the UNIQUAC interaction parameter between species i and j. 2.4. Ion Association. The equilibrium of generic salt MxXm in aqueous solution and its corresponding stability constant, xXm KM , are: S M xX m ↔ x M m + + m X x − KSMxX m

[MxX m] = m+ x [M ]f [X x −]mf

[H+]f =

22.42 34.20 32.97

(32)

[H] t 1 + KSHCl[Cl−]f

[Mg 2 +]f =

(33)

[Ca] t +

1 + KSCaCl [Cl−]f

(34)

[Mg] t +

1 + KSMgCl [Cl−]f

(35)

[Cl−]f = [Cl] t/(1 + KSKCl[K+]f + KSNaCl[Na +]f +

+ KSHCl[H+]f + KSCaCl [Ca 2 +]f +

+ KSMgCl [Mg 2 +])

(36)

All free concentrations can be now calculated from eqs 31 to 36, once the total concentrations and stability constants are known. Kester and Pytkowicz43 expressed the stability constants as a function of the solution ionic strength by: ln(KSMxX m) = A′ + B′I

(37)

xXm KM S

values or correlations are available in the literature for several salts and complexes.43,52−56 In this work, this approach has been implemented whenever possible. 2.5. Determination of KBA and Parameters of γi̅ . The simultaneous fitting of equilibrium constant and parameters of activity coefficient models (for the solid phase) to experimental data usually gives rise to an undesirable degree of correlation between the two sets of parameters.8,25 Nevertheless, the application of thermodynamically consistent models that provide estimates of equilibrium constants independent of the activity coefficients in the sorbed phase could avoid such correlation.8,25 Two powerful approaches in the literature are due to Gaines and Thomas8,45 and Ioannidis et al.8,28 The fundamental equation of Gaines and Thomas can be applied when the

(30)

[K] t 1 + KSKCl[Cl−]f

γi̅ = γi = 1

7.41 16.02 15.12

1 + KSNaCl[Cl−]f

(28)

where [K]t is the total concentration of potassium present in solution in whatever form. Taking into account the stability constant, [KCl] = KSKCl [K+]f[Cl−]f, and eq 30 the concentration of free K+ can be obtained after rearrangement: [K+]f =

UNIQUAC

[Na] t

[Ca 2 +]f =

(29)

ideal model

3.41 11.35 10.52

[Na +]f =

where [Mm+]f and [Mx−]f are the free concentrations of cation Mm+ and anion Xx− in solution. As mentioned above, some of the ions are not available for ion exchange since they are not present as free species but rather as ion pairs. The method of Kester and Pytkowicz43 can be successfully applied to determine the real ionic concentrations in solution, for instance, in the case of the five-component system studied in this work and experimentally measured by Vo and Shallcross, where K+, Na+, H+, Ca2+, and Mg2+ are present with co-ion Cl− in solution. Some of the particles will exist as KCl, NaCl, MgCl+, and CaCl+; it is assumed that all CaCl2 and MgCl2 dissociate to CaCl+ and Cl−, and MgCl+ and Cl−, respectively. The material balance to potassium is: [K]t = [K+]f + [KCl]

model for γi̅ NRTL

(31)

The same approach may be applied to the remaining ionic species, giving rise to: 1771



Article

Table 4. Calculated Results for the Predicted Multicomponent Ion Exchange Isotherms of Database (Table 1) range of ionic strength system

mol·kg−1

K+/Na+/H+/Cl−/Amberjet 1200H

0.10

K+/Na+/Ca2+/Cl−/Amberjet 1200H

0.10

K+/Na+/Mg2+/Cl−/Amberjet 1200H

0.10

Na+/H+/Ca2+/Cl−/Amberjet 1200H

0.10

Na+/H+/Mg2+/Cl−/Amberjet 1200H

0.10

Na+/Mg2+/Ca2+/Cl−/Amberjet 1200H

0.10

K+/H+/Ca2+/Cl−/Amberjet 1200H

0.10

K+/H+/Mg2+/Cl−/Amberjet 1200H

0.10

K+/Mg2+/Ca2+/Cl−/Amberjet 1200H

0.10

H+/Mg2+/Ca2+/Cl−/Amberjet 1200H

0.10

K+/Na+/Ca2+/Cl−/Clinoptilolite

0.1

Ca2+/Mg2+/Na+/Cl−/Amberlite 252

0.1

Na+/Cs+/Mn2+/Cl−/Dowex 50W-X8

0.1

Cd2+/Zn2+/H+/NO3−/Amberlite IR 120

0.1

K+/Na+/Ca2+/Mg2+/Cl−/Amberjet 1200H

0.10

K+/Na+/H+/Ca2+/Cl−/Amberjet 1200H

0.10

K+/Na+/H+/Mg2+/Cl−/Amberjet 1200H

0.10

K+/H+/Ca2+/Mg2+/Cl−/Amberjet 1200H

0.10

Na+/H+/Ca2+/Mg2+/Cl−/Amberjet 1200H

0.10

model for γi̅ Ternary Systems 1.00 Wilson NRTL UNIQUAC 1.50 Wilson NRTL UNIQUAC 1.50 Wilson NRTL UNIQUAC 1.50 Wilson NRTL UNIQUAC 1.50 Wilson NRTL UNIQUAC 1.50 Wilson NRTL UNIQUAC 1.50 Wilson NRTL UNIQUAC 1.50 Wilson NRTL UNIQUAC 1.50 Wilson NRTL UNIQUAC 1.50 Wilson NRTL UNIQUAC 0.15 Wilson NRTL UNIQUAC 0.15 Wilson NRTL UNIQUAC 0.15 Wilson NRTL UNIQUAC 0.15 Wilson NRTL UNIQUAC Quaternary Systems 1.50 Wilson NRTL UNIQUAC 1.50 Wilson NRTL UNIQUAC 1.50 Wilson NRTL UNIQUAC 1.50 Wilson NRTL UNIQUAC 1.50 Wilson NRTL 1772

range of γi 0.5857

0.8227

0.2722

0.7749

0.2881

0.7747

0.2525

0.8126

0.2837

0.8113

0.1305

0.7692

0.2516

0.8170

0.2819

0.8167

0.1300

0.7588

0.1296

0.7947

0.4010

0.8150

0.2421

0.7755

0.3351

0.7866

0.3594

0.7865

0.1921

0.7667

0.2707

0.8193

0.2873

0.8194

0.1754

0.8131

0.1764

0.8083

range of γi̅

AARD/%

0.5232 0.5002 0.0432 0.2654 0.2061 0.1794 0.2071 0.3510 0.2180 0.3285 0.2590 0.0948 0.0905 0.4062 0.0374 0.4091 0.4425 0.4073 0.5729 0.5763 0.0459 0.4720 0.6086 0.0492 0.5002 0.4830 0.4488 0.4936 0.4731 0.1273 0.0602 0.0755 0.0470 0.3813 0.3887 0.2336 0.3651 0.3628 0.4462 0.3181 0.3501 0.0494

1.0080 1.0204 1.0032 1.0554 1.0516 1.2515 0.9888 1.3004 1.1371 1.0197 1.0156 0.9651 1.0285 1.3279 1.1835 1.0162 1.2410 0.9764 0.9960 0.9942 1.0878 0.9835 0.9871 1.0157 0.9894 0.9863 1.0722 0.9919 0.9877 0.9786 1.0000 1.0000 1.0000 1.9030 1.8012 1.8036 0.9458 0.9532 0.9242 1.3775 1.3554 1.2548

2.99 3.13 5.47 25.03 30.61 43.04 10.81 12.26 23.41 25.73 27.60 26.36 19.80 10.84 22.44 2.93 4.16 3.83 17.69 18.38 19.65 13.86 7.14 16.68 2.86 3.17 4.05 1.65 1.76 3.47 57.20 54.31 54.70 13.28 12.75 20.01 57.73 54.51 56.34 9.83 9.95 6.84

0.2597 0.4334 0.3761 0.3008 0.2120 0.0567 0.2010 0.4487 0.0615 0.3124 0.3334 0.0502 0.2400 0.2440

1.0041 1.0240 1.0691 1.0405 1.0398 1.0986 0.9939 1.1897 1.1604 0.9758 1.0007 1.0424 0.9599 1.6579

3.01 3.53 6.45 11.08 13.06 15.66 8.82 11.66 27.85 3.41 3.02 4.91 3.02 4.82

AARD ideal/% 9.92

36.55

16.37

31.34

36.66

68.64

42.00

28.67

68.86

67.14

58.42

23.08

69.70

39.41

22.14

17.45

24.01

27.54

23.33



Article

Table 4. continued range of ionic strength mol·kg−1

system

K+/Na+/H+/Ca2+/Mg2+/Cl−/ Amberjet 1200H

model for γi̅ Quaternary Systems UNIQUAC Quinary System 1.50 Wilson NRTL UNIQUAC

0.10

logarithm of the corrected selectivity coefficient, ln KAaB, spans the entire range of the cation equivalent ionic fraction in the solid, Yi∈[0,1], which is a very limiting condition. Note the difference between the mole fraction, yi, and the equivalent ionic fraction, Yi, whose relation between them involves all charged species (n): yzi Yi = n i ∑ j = 1 yj zj (38)

ln

γ A̅ z B(θ1)

+ ln

γB̅ zA(θ1) γB̅ zA(θ2)

= ln

KaAB(θ1)

− ln

KaAB(θ2)

0.2039

0.8157

range of γi̅

AARD/%

0.0941

1.8366

7.61

0.2844 0.2316 0.0553

0.9954 1.1869 1.0936

5.78 4.74 7.80

AARD ideal/%

30.64

(Wilson, NRTL, and UNIQUAC) necessary for the prediction of the multicomponent isotherms. Then, the multicomponent equilibria results are analyzed in detail, taking into account the accuracy of the predictions achieved for ternary, quaternary, and quinary systems. The capability of the NRTL and UNIQUAC models to represent the nonideal behavior of the ion exchanger is compared with the Wilson equation, which is almost uniquely adopted in the literature. The influence of the quality of the binary data upon multicomponent predictions is pointed out, as well as the extension of the ion association. To quantify the accuracies of correlation and predictions, we use in this work the average absolute relative deviation (AARD), since it provides clear and quantitative information:

Alternatively, the method developed by Ioannidis et al.8,28 adopted in this work avoids this constraint as well as the significant errors inherent to the extrapolations needed in boundaries of the Yi interval. The main equation is: γ A̅ z B(θ2)

range of γi

(39)

AARD =

From a set of experimental data one chooses one reference point (θ1) and calculates ln KAaB for all of the remaining points (θ2). Then, the differences ln KAaB(θ1) − ln KAaB(θ2) are computed, and the parameters of the activity coefficient model adopted for γi̅ are optimized by minimizing the sum of residuals corresponding to eq 39.

⎡ NDP n ∑ j = 1 ⎢∑i =c 1 ⎢⎣

yiexp − yicalc

NDP ·nc

yiexp

⎤ ⎥ ⎦ j⎥ (40)

where superscripts exp and calc stand for experimental and calculated mole fractions of counterion i, respectively, and index j span all points of equilibrium data (NDP values). Almost all publications report average absolute deviations or average squared deviations, which give rise to much smaller errors. Nonetheless, this approach is inadequate to quantify particularly the dilute region of isotherms, and then it was not selected here. 4.1. Brief Analysis of Binary Isotherms. In Table 2 the detailed results obtained for the constitutive binary systems are compiled. It contains the identification (counterions, co-ion, and ion exchanger) and total equivalent concentration (Ct) of each system, the maximum variation of the ionic strength that can occur, the models utilized for the activity coefficients inside exchanger (γi̅ = NRTL, UNIQUAC, WILSON), the equilibrium constants and γi̅ parameters fitted to the experimental data (Λ12 and Λ21 for Wilson, Δg12 = g12 − g22 and Δg21 = g21 − g11 for NRTL, and a12 and a21 for UNIQUAC), and finally the average absolute relative deviations. According to the computed global deviations listed in Table 3 it is possible to conclude that both Wilson and NRTL models provide accurate representations of ion exchange equilibrium (AARDWilson = 3.21 % and AARDNRTL = 3.41 %), whereas UNIQUAC performs worse with more than the double of the deviation (AARDUNIQUAC = 7.41 %). These results corroborate our previous findings.19 The nonidealities are extremely important for correct modeling of these binary isotherms, since the global error for the ideal mass action law (γi = γi̅ = 1.0) attains 22.42 %. The determination of KAB and γi̅ parameters was accomplished by Ioannidis et al.28 approach since the majority of systems do not span all equivalent mole fraction range in the solid (YA), which happened with 19 of the 22 binary systems present in Table 2.

3. DATABASE In Table 1 the ion exchange systems studied in this work are listed. The most important information is compiled, namely, the system (counterions, co-ion, and ion exchanger), number of data points, NDP, total equivalent concentration, Ct(N), and references; the temperature is 298.15 K. Data comprehend 14 ternary systems, 5 quaternary systems, 1 quinary system, and the 22 constitutive binary systems embodied in the previous multicomponent mixtures. The binaries include 1−1, 1−2, and 2−2 charge pairs of counterions. In the particular case of multicomponent mixtures, which are the main objective of this Article, there are 16 systems which possess experimental isotherms for several total equivalent concentrations ((0.1, 0.2, 0.5, and 1.0) N). All parameters necessary for the calculations are given in Appendix: the Pitzer parameters for the activity coefficients of ions in solution are collected in Tables A1 and A2; the structural parameters of the UNIQUAC model for the activity coefficients of cations in the exchanger phase are listed in Table A3; the stability constants or Kester and Pytkowicz parameters (eq 37) for the ion association calculations are given in Table A4. 4. RESULTS AND DISCUSSION In this section the calculated results are presented and discussed. One starts with the constitutive binary systems used to fit the equilibrium constants and binary parameters of γi̅ 1773



Article

4.2. Prediction of Multicomponent Isotherms. In Table 4 the average deviations obtained for the 20 multicomponent systems are listed along with the ranges of ionic strength, γA (solution) and γA̅ (exchanger), and the errors for the ideal model approach. The predictions were accomplished using KAB and γi̅ parameters fitted before for each counterion pair. The global AARDs found (see Table 2) show that the mass action law with NRTL equation for γi̅ achieves better predictions, followed by the Wilson and UNIQUAC alternatives: AARDWilson = 11.53 %, AARDNRTL = 11.35 %, and AARDUNIQUAC = 16.02 %. The grand AARDs (i.e., binary and multicomponent) follow the same trend, respectively, 10.66 %, 10.52 %, and 15.12 %, it being possible to conclude that both NRTL and Wilson models may be adopted to represent the nonideal behavior of the ion exchanger, while UNIQUAC is not recommended. This finding corroborates that for binaries analyzed in the previous section. In Figure 1a the experimental data and calculated isotherms for the 10 ternary systems published by Vo and Shallcross29 are

multicomponent systems of Vo and Shallcross containing four and five counterions (see Tables 1 and 4) are likewise graphed in Figure 1b and point out once again that UNIQUAC expressions are consistently responsible for underpredicted values. In Figure 2 the results obtained for the last four ternary systems of Table 2 are shown (they have been separated from

Figure 2. Predicted versus experimental ion exchange equilibrium data for the last four ternary systems of database. Model: mass action law; ion association; γi by Pitzer, and γi̅ computed by ○, Wilson; ×, NRTL; and +, UNIQUAC equations.

Figure 1a, since all data in Figure 1a and b were taken from the same reference, hence they are naturally uniform). The significant dispersion of data and huge underestimations are essentially due to the systems K+/Na+/Ca2+/Cl−/clinoptilolite and Na+/Cs+/Mn2+/Cl−/Dowex 50W-X8, for which the deviations found are (57.20 to 54.70) % and (54.51 to 57.73) %, respectively. In these cases, the Wilson, NRTL, and UNIQUAC expressions behave not only poorly but also similarly. Such large deviations may rely on the KAB and γi̅ parameters provided by the constitutive binary systems Na+/ Ca2+, K+/Ca2+, and Na+/Mn2+, since the calculated errors for these pairs are large (c.a., 6 to 16 %; see Table 2). The relevance of the nonidealities in both the solid and liquid domains are confirmed by the activity coefficients γi̅ and γi, respectively, presented in Table 4. Their intervals of variation demonstrate that all systems suffer the consequences of nonidealities: in solution, γi deviates undoubtedly from unity, and in the exchanger phase the minimum values of γi̅ are markedly nonunitary. It is worth noting that most ionic strengths attain 1.5 molal, which exceeds the validity range of the well-known and simple Debye−Hückel equation and justifies the selection of the Pitzer model in this essay. The large errors offered by eq 9 (ideal model) in contrast to those computed by mass action law with Wilson+Pitzer, NRTL +Pitzer, and UNIQUAC+Pitzer combinations highlights the importance of the activity coefficients in the multicomponent predictions: AARDWilson = 11.53 %, AARDNRTL = 11.35 %, and AARDUNIQUAC = 16.02 % against AARDideal = 34.20 % (see Table 3). In Figure 3 the experimental versus ideally calculated molar fractions of all multicomponent systems of database are plotted to emphasize the deficient performance of the ideal modeling approach. In contrast to Figures 1a, 1b, and 2, the enormous scattering found here confirms the activity coefficients are absolutely necessary.

Figure 1. Predicted versus experimental ion exchange equilibrium data of Vo and Shallcross29 (Table 1). Model: mass action law; ion association; γi by Pitzer, and γi̅ computed by ○, Wilson; ×, NRTL; and +, UNIQUAC equations. (a) Data for 10 ternary systems; (b) data for 5 quaternary and 1 quinary systems.

plotted. Most points are essentially distributed around the diagonal, but one may observe that the UNIQUAC model exhibits underpredictions in the range of 0.1 < yi < 0.7. The six 1774



Article

the percentage of free Ca2+, Mg2+, and K+ drop from c.a. 79, 84, and 95 % to 38, 40, and 75 %, respectively. As a final illustration of the reliable predictions achieved by the ion exchange model proposed in this work with NRTL equations for γi̅ , it is given in Figure 5 triangular plots of

Figure 3. Predicted (ideal model) versus experimental ion exchange equilibrium data for all multicomponent systems of database (Table 1).

The ion association was also taken into account in this work by implementing the method of Kester and Pytkowicz. For illustration, the influence of the partial dissociation of salts or complex formation in solution is shown in Figure 4 for the

Figure 5. Experimental (●) and predicted (○) isotherms for the ternary system Na+/Mg2+/Ca2+/Cl−/Amberjet 1200H of database (Table 1) for total concentrations of 0.1, 0.2, 0.5, and 1.0 N. Model: mass action law; ion association; by Pitzer, and γi̅ computed by NRTL equations.

isotherms of Na+/Mg2+/Ca2+/Cl−/Amberjet 1200H for 0.1, 0.2, 0.5, and 1.0 N. The figure shows that both experimental and calculated data exhibit the same trend, being very near to each other and, in a large number, almost overlapped. In Figure 6

Figure 4. Variation of the percentage of free counterions in solution as a function of total concentration for the quinary system K+/Na+/H+/ Ca2+/Mg2+/Cl−/Amberjet 1200H (data from Vo and Shallcross29). The mole fractions in solution are fixed and equal [0.40, 0.20, 0.20, 0.15, 0.05].

quinary ion exchange equilibria K+/Na+/H+/Ca2+/Mg2+/Cl−/ Amberjet 1200H. It indubitably points out that the percentage of free counterions decreases when total normality of solution increases, because the ions in solution get progressively closer to each other. The data plotted span the range Ct∈[0.1,1.0] N and refer to the following global cation mole fractions (i.e., free plus associated cations): xK+ = 0.40, xNa+ = 0.20, xH+ = 0.20, xCa2+ = 0.15, and xMg2+ = 0.05. It is worth noting the mole fraction vector must be fixed to evaluate exclusively the effect of total concentration. The notable changes of free cation fractions indicate that it is fundamental to include the association phenomenum in the ion exchange equilibrium modeling. For instance,

Figure 6. Isotherms of the constitutive binary systems of Na+/Mg2+/ Ca2+/Cl−/Amberjet 1200H: ▲, Mg2+/Na+/Cl−; ■, Ca2+/Na+/Cl−, and ●, Ca2+/Mg2+/Cl−. Model: mass action law; ion association; γi by Pitzer, and γi̅ computed by , Wilson; −−, NRTL; ---, UNIQUAC (data from Vo and Shallcross29).

the three constitutive binary isotherms are drawn along with experimental data, namely, Mg2+/Na+, Ca2+/Na+, and Ca2+/ Mg2+ for 0.1 N. The accurate results achieved by mass action law with the Pitzer equation (γi) combined with NRTL or Wilson or even with UNIQUAC models (γi̅ ) are also obvious from the good fittings accomplished. 1775



Article

5. CONCLUSIONS The ion exchange equilibrium of binary and multicomponent systems was studied under the scope of the mass action law. The nonidealities in both solution and solid particles were taken into account, along with ion association phenomena in the liquid phase. The Pitzer model was utilized for the activity coefficients in the liquid, whereas the nonideal behavior of the sorbed multicomponent phase was studied for the first time with NRTL and UNIQUAC models. The mass action law provided good results with NRTL equations, which were very similar to those accomplished by the Wilson expressions. On the contrary, the UNIQUAC alternative performed poorer. The average absolute relative deviations for the correlation of 22 constitutive binary systems (327 data points) were 3.21 %, 3.41 %, and 7.41 % using Wilson, NRTL, and UNIQUAC models, respectively. The pure prediction of 20 multicomponent systems (14 ternary, 5 quaternary, and 1 quinary mixtures, totalizing 1167 data points) was accomplished with 11.53 %, 11.35 %, and 16.02 %, respectively. In the whole, this work clearly shows that both NRTL and Wilson equations can be used reliably to predict multicomponent ion exchange equilibrium. The ion association in solution was generally significant, and results undoubtedly highlighted that the percentage of free counterions decreases with increasing total normality. The chief importance of accurate equations for the nonideal behavior of ion exchange system was definitely emphasized by the large global deviation achieved with the ideal model (γi̅ = γi = 1), AARDideal = 32.97.

Table A3. Structural Parameters for the UNIQUAC Equation60,61

MgCl KCl NaCl CaCl+

■

system

kg·mol

MgCl2 KCl NaCl CaCl2 HCl MnCl2 CsCl HNO3 Cd(NO3)2 Zn(NO3)2

kg·mol

0.35235 0.04835 0.0765 0.3159 0.1775 0.3272 0.0300 0.1119 0.2865 0.3481

1.6815 0.2122 0.2664 1.6140 0.2945 1.5502 0.0558 0.3206 1.6680 1.6913

system +

2+

kg·mol −

Na /Ca /Cl K+/Ca2+/Cl− Mg2+/Ca2+/Cl− K+/Na+/Cl− K+/Mg2+/Cl− Na+/Mg2+/Cl−

0.07 0.032 0.07 −0.012 0 0.07

0.651 −0.491 −0.537 1.073

−0.011 −0.464 −1.002 −0.442

AUTHOR INFORMATION

Funding

J.P.S.A. would like to acknowledge the funding from the European Community's Seventh Framework Programme FP7/ 2007−2013 under grant agreement No. CP-IP 228589-2 AFORE. P.F.L. wishes to acknowledge the grant provided by Fundaçaõ para a Ciência e Tecnologia (Portugal; SFRH/BPD/ 63214/2009). The authors thank Pest-C/CTM/LA0011/2011 for CICECO funding. Notes

The authors declare no competing financial interest.

■

NOMENCLATURE Aγ Debye−Huckel constant (kg0.5·mol−0.5) AzA,BzB,CzC Counterions with valences zA,zB,zC A̅ zA,B̅ zB Counterions with valences zA,zB inside the exchanger ai Activity of species i in solution ai̅ Activity of species i in exchanger aij UNIQUAC interaction parameter of i−j pair (K) Azi i Generic counterion i with valence zi AARD Average absolute relative deviation Bij Second virial coefficient (kg·mol−1) B′ij Variable in the Pitzer equation (kg2·mol−2) (0) Cij and Cij Parameters in the Pitzer equation (kg2·mol−2) e 1.60206·10−19 C, electron charge F Debye−Hückel parameter gij Energy parameter characteristic of the i−j interaction (J·mol−1) I Ionic strength (mol·kg−1) k 1.38066·10−23 J·K−1, Boltzmann's constant KAaB Corrected selectivity coefficient KAB, KCB , KAC Thermodynamic (equilibrium) constant Kc Selectivity coefficient (molal basis) xMm KM Stability constant (molal basis) S Mm+ Cation mi Molality of species i (mol·kg−1) mt Total molality of ionic species (mol·kg−1)

kg ·mol−2 2

0.00519 −0.00084 0.00127 −0.00034 0.00080 −0.02050 0.00038 0.00100 −0.02565 −0.01567

Ψi,j,k −1

B′/(kg·mol−1)

*E-mail: [email protected].

Table A2. Binary Salt Parameters of the Pitzer Equation for the Systems under Study47 Θi,j

A′

Corresponding Author

C(0) ij −1

0.307 0.028 0.162 0.596 0.326 0.217 0.282 0.176 0.905

+

Table A1. Single Parameters of the Pitzer Equation for the Systems under Study47 β(1) ij

q

0.167 0.005 0.064 0.452 0.182 0.099 0.147 0.073 0.845

ion pair

APPENDIX In Tables A1 and A2 the Pitzer parameters for the activity coefficients of ions in solution are collected. The structural parameters of the UNIQUAC model for the activity coefficients of cations in exchanger are given in Table A3. The stability constants or Kester and Pytkowicz parameters are compiled in Table A4.

−1

r

Ca2+ H+ Mg2+ K+ Na+ Mn2+ Cd2+ Zn2+ Cs+

Table A4. Kester and Pytkowicz Parameters (Equation 37)43

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β(0) ij

cation

kg ·mol−2 2

−0.014 −0.025 −0.012 −0.0018 −0.022 −0.012 1776



Article

6.022·1023 mol−1, Avogadro’s constant Number of counterions Molecular surface parameter of UNIQUAC model Molecular size parameter of UNIQUAC model 8.3145 J·mol−1·K−1, universal gas constant Absolute temperature (K) Pure-component molar volume (cm3·mol−1) Mole fraction of species i in solution Anion Mole ionic fraction of species i in exchanger Equivalent ionic fraction of species i in exchanger Coordination number of UNIQUAC model Valence of ionic species i

N0 nc qi ri R T vi xi Xx− yi Yi z zi

(9) Rodriguez, J. F.; Valverde, J. L.; Rodrigues, A. E. Measurements of effective self-diffusion coefficients in a gel-type cation exchanger by the zero-length-column method. Ind. Eng. Chem. Res. 1998, 37 (5), 2020−2028. (10) Valverde, J. L.; de Lucas, A.; Carmona, M.; Gonzalez, M.; Rodriguez, J. F. A generalized model for the measurement of effective diffusion coefficients of heterovalent ions in ion exchangers by the zero-length column method. Chem. Eng. Sci. 2004, 59 (1), 71−79. (11) Silva, C. M.; Lito, P. F. Application of the Maxwell-Stefan approach to ion exchange in microporous materials. Batch process modelling. Chem. Eng. Sci. 2007, 62 (23), 6939−6946. (12) Ferreira, T. R.; Lopes, C. B.; Lito, P. F.; Otero, M.; Lin, Z.; Rocha, J.; Pereira, E.; Silva, C. M.; Duarte, A. Cadmium(II) removal from aqueous solution using microporous titanosilicate ETS-4. Chem. Eng. J. 2009, 147 (2−3), 173−179. (13) Camarinha, E. D.; Lito, P. F.; Antunes, B. M.; Otero, M.; Lin, Z.; Rocha, J.; Pereira, E.; Duarte, A. C.; Silva, C. M. Cadmium(II) removal from aqueous solution using microporous titanosilicate ETS-10. Chem. Eng. J. 2009, 155 (1−2), 108−114. (14) Barreira, L. D.; Lito, P. F.; Antunes, B. M.; Otero, M.; Lin, Z.; Rocha, J.; Pereira, E.; Duarte, A. C.; Silva, C. M. Effect of pH on cadmium (II) removal from aqueous solution using titanosilicate ETS4. Chem. Eng. J. 2009, 155 (3), 728−735. (15) Lito, P. F.; Silva, C. M. Comparison between Maxwell-Stefan and Nernst-Planck equations to describe ion exchange in microporous materials. Diffusion in Solids and Liquids III 2008, 273−276, 776−781, 837. (16) Lito, P. F.; Aniceto, J. P. S.; Silva, C. M. Maxwell-Stefan Based Modelling of the Ion Exchange of Systems Containing Common Species and Distinct Sorbents. Int. J. Environ. Sci. Technol. 2012, submitted. (17) Wesselingh, J. A.; Vonk, P.; Kraaijeveld, G. Exploring the Maxwell-Stefan description of ion-exchange. Chem. Eng. J. Biochem. Eng. 1995, 57 (2), 75−89. (18) Melis, S.; Cao, G.; Morbidelli, M. A New Model for the Simulation of Ion Exchange Equilibria. Ind. Eng. Chem. Res. 1995, 34 (11), 3916−3924. (19) Aniceto, J. P. S.; Cardoso, S. P.; Faria, T. L.; Lito, P. F.; Silva, C. M. Modeling ion exchange equilibrium: Analysis of exchanger phase non-ideality. Desalination 2012, 290 (0), 43−53. (20) Dranoff, J.; Lapidus, L. Equilibrium in Ternary Ion Exchange Systems. Ind. Eng. Chem. 1957, 49 (8), 1297−1302. (21) Pieroni, L. J.; Dranoff, J. S. Ion exchange equilibria in a ternary system. AIChE J. 1963, 9 (1), 42−45. (22) Klein, G.; Tondeur, D.; Vermeulen, T. Multicomponent Ion Exchange in Fixed Beds. General Properties of Equilibrium Systems. Ind. Eng. Chem. Fundam. 1967, 6 (3), 339−351. (23) Smith, R. P.; Woodburn, E. T. Prediction of multicomponent ion exchange equilibria for the ternary system SO42−-NO3−-Cl from data of binary systems. AIChE J. 1978, 24 (4), 577−587. (24) Sengupta, M.; Paul, T. B. Multicomponent ion exchange equilibria. I. Zn2+-Cd2+-H+ and Cu2+-Ag+-H+ on Amberlite IR 120. React. Polym., Ion Exch., Sorb. 1985, 3 (3), 217−229. (25) Shallcross, D. C.; Herrmann, C. C.; McCoy, B. J. An improved model for the prediction of multicomponent ion exchange equilibria. Chem. Eng. Sci. 1988, 43 (2), 279−288. (26) Martínez, A. d. L.; Cañizares, P.; Díaz, J. Z. Binary ion exchange equilibrium for Ca2+, Mg2+, K+, Na+ and H+ ions on Amberlite IR-120. Chem. Eng. Technol. 1993, 16 (1), 35−39. (27) Mehablia, M. A.; Shallcross, D. C.; Stevens, G. W. Prediction of multicomponent ion exchange equilibria. Chem. Eng. Sci. 1994, 49 (14), 2277−2286. (28) Ioannidis, S.; Anderko, A.; Sanders, S. J. Internally consistent representation of binary ion exchange equilibria. Chem. Eng. Sci. 2000, 55 (14), 2687−2698. (29) Vo, B. S.; Shallcross, D. C. Ion Exchange Equilibria Data for Systems Involving H+, Na+, K+, Mg2+, and Ca2+ Ions. J. Chem. Eng. Data 2005, 50 (3), 1018−1029.

Subscripts

a c f t

Anion Cation Free Total

Superscripts

calc Calculated value exp Experimental value − (top bar) identifies exchanger phase Greek Letters

αij

NRTL parameter related to the nonrandomness in the mixture β(0) Parameter in the Pitzer equation (kg·mol−1) ij Parameter in the Pitzer equation (kg·mol−1) β(1) ij Δgij gij − gjj (J·mol−1), interaction parameter of i−j pair in the NRTL model ε Dielectric constant (C2·N−1·m−2) ϕi Molecular volume fraction of species i of UNIQUAC model γi Activity coefficient of species i in solution γi̅ Activity coefficient of species i in exchanger λi Distribution coefficient of species i Λij,Λji Wilson parameters θi Surface fraction of species i of UNIQUAC model Θi,j Parameter in the Pitzer equation (kg·mol−1) ρw Density of pure solvent (g·cm−3) Ψ Third virial coefficient (kg2·mol−2)

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