Temperature-Dependent Model for the Prediction of Binary Ion

Article pubs.acs.org/IECR

Temperature-Dependent Model for the Prediction of Binary Ion Exchange Equilibria Involving Na+, K+, Ca2+ and Mg2+ Ions Masooma Rustam and David C. Shallcross* Department of Chemical and Biomolecular Engineering University of Melbourne, Melbourne, Victoria 3010, Australia ABSTRACT: The effect of temperature on the ion exchange equilibrium behavior of binary systems is studied experimentally and modeled. The study considers the exchange between binary systems involving Na+, K+, Ca2+ and Mg2+ ions with Cl− as a nonexchanging anion on Rohm and Hass gel-type Amberjet 1200H resin as the ion exchange medium. Experimental equilibrium data is obtained for the six constitutive binary systems at 4.0, 25.0 and 60.0 °C with a total solution concentration of 0.10 N. The nonidealities in the solution and exchanger phases are modeled by applying the Pitzer and Wilson models, respectively. Thermodynamic equilibrium constant and Wilson interaction parameters are modeled with respect to temperature. The model proposed allows the influence of temperature on the ion exchange equilibrium behavior to be predicted well.

1. INTRODUCTION Prediction and modeling of ion exchange equilibria are essential for the design and development of efficient ion exchange processes. Physicochemical parameters such as solvent composition, total solution concentration and system temperature influence the selectivity of ion exchange materials for particular counterions.1 Although ion exchange has been extensively studied for a long time, the role of temperature continues to be largely underestimated because of the general assumption that temperature has an insignificant effect on all ion exchange systems. A broad survey of the published literature has shown that many have investigated the effect of temperature on ion exchange equilibria experimentally.2−17 They studied temperature dependence of ion exchange on various types of exchangers and different cations involved. They concluded their findings in terms of the evaluation of thermodynamic properties such as ΔF°, ΔH° and ΔS°. None of them, however, model the effect of temperature on the ion exchange equilibria to predict the ion exchange equilibria. All studied binary systems only, except for Barros et al.,3−5 who studied ternary systems as well. Semitheoretically approaches to investigate the role of temperature on ion exchange equilibria have been investigated by a number of researchers.18−26 Barri et al.18 investigated the binary and ternary cation exchange in zeolites involving Na+, Ca2+ and Mg2+ at 25 and 65 °C. They fitted the corrected selectivity coefficient (Kc) data to polynomial equations and then from these equations, thermodynamic parameters were calculated. It is found that isotherms at various temperatures cannot be calculated from Ka because the activity coefficients of the two ions in the zeolite phase must change with temperatures. They made an attempt to predict ternary exchange data from corresponding binary isotherms. de Lucas et al.20 also studied the binary ion exchange equilibrium of calcium, magnesium, potassium, sodium and hydrogen ions onto Amberlite IR-120 at three temperatures 10, 30 and 50 °C. They characterized the equilibrium based on the law of mass action using the Wilson model27 and the Meissner and Kusik model28 to determine the activity coefficients in the © XXXX American Chemical Society

resin and the solution phases. They observed the increased selectivity with the charge on the cation and with decreasing volume of the hydrate ion. They also observed the increased preference of one cation over the other with increases in the temperature. Mehablia et al.21 studied the effect of temperature on a binary system (Ca−K) at 6, 21 and 65 °C. They considered the nonidealities in the solution phase by applying the Pitzer’s model29 and also incorporated the ion-association theory. Wilson’s model27 was applied to consider the nonidealities in the exchanger phase. It was observed that calcium preference increased by increasing the temperature and binary interaction parameters obtained were nearly constant over the range of temperature studied. They observed a linearity of the equilibrium constant and the binary interaction parameters with respect to temperature and the predicted isotherms fitted the experimental data well. Valverde et al.23,24 studied the equilibrium behavior of H+/ Cd2+, H+/Cu2+, H+/Zn2+, Na+/Cd2+, Na+/Cu2+ and Na+/Zn2+ on a strong acid resin, Amberlite IR-120 at 10 and 30 °C. They correlated the experimental equilibrium data using the homogeneous mass action law model. They observed that the temperature of the reaction has a strong influence on the equilibrium behavior. The three unknown parameters (the equilibrium constant, KAB and Wilson interaction parameters) were obtained by fitting the experimental data to the model. The affinity of the resin for the metal ions increased with increasing temperature. Fridriksson et al.25 investigated the ion exchange equilibrium between heulandites and aqueous solutions for binary systems involving Ca2+, K+, Na+ and Sr2+. All binary systems such as Ca2+−K+, Ca2+−Na+, K+−Na+, K+−Sr2+, Na+−Sr2+and Ca2+− Sr2+ were investigated at 55 and 85 °C. Experimentally determined isotherms were used to derive the equilibrium Received: May 17, 2014 Revised: July 29, 2014 Accepted: August 8, 2014

A

dx.doi.org/10.1021/ie5020102 | Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Industrial & Engineering Chemistry Research

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2.1. Solution Phase Nonidealities. Due to incomplete dissociation of the ions not all the ions are available for ion exchange. As well as it being necessary to model the equilibrium between the ions in the solution and exchanger phases, it is also necessary to model the equilibrium between the ions and the ion pairs. Both sets of equilibria are dependent on temperature. Equilibrium will be established between the associated and dissociated forms corresponding to the equation:

constant for the ion exchange reaction and asymmetric Margules models describing the extent of nonideality in extra framwork solid solutions in heulandites. The applicability of the present experimental results and the thermodynamic models was assessed by calculating the composition of heulandites in Icelandic geothermal systems from known compositions using the regressed thermodynamic properties of Ca−Na exchange at 85 °C. Mumford et al.26 investigated the exchange of Cu2+, Na+ and + H with Amberlite IRC 748 at temperatures of 4, 12, 20 and 40 °C using Cl− as the nonexchanging anion by using a twoparameter temperature-dependent, semiempirical thermodynamic ion exchange model to describe binary systems. The semiempirical thermodynamic ion exchange model was shown to predict the shape of the equilibrium ion exchange curves accurately, despite the low concentrations of copper used in this study (0.001 57−0.007 87 N). There was a consistent increase in selectivity of the resin toward copper with an increase in temperature. Ion exchange equilibrium is a temperature-dependent condition and temperature can influence the preference that the exchange phase displays for one ion over another. Several methods are available for the prediction of ion exchange equilibria, but there is a knowledge gap in describing the temperature dependence of ion exchange equilibria for multicomponent systems. This current work explores how a well-understood model to predict multicomponent ion exchange equilibrium behavior at 25 °C may be extended to account for the variations in equilibrium behavior as temperature is varied. In this work, an existing model is adapted to consider the influence of temperature on its model parameters. Extensive experimental data obtained for the six constitutive binary systems of Na+, K+, Ca2+ and Mg2+ ions at three different temperatures allows the effect of temperature to be properly incorporated into the existing models. A model that incorporates temperature effects will permit the design of more efficient ion exchange processes by allowing designers to optimize the operating temperature of the process. The model will also allow workers to use experimental data collected at one temperature to predict performance at another temperature. The novelty of the work is that it draws on an extensive and consistent set of experimental binary equilibrium data for six binary systems at three different temperatures. Other data sets available in the published literature are not as extensive, either being limited to just a single binary system or to only a couple of different temperatures.

M xX m ⇌ x M m + + m X x −

The method of Kester and Pykowicz was used for this purpose. In this method, the stability constant KMX S is defined as

⎛ α̲ A ⎞ z B ⎛ αB ⎞ zA =⎜ ⎟ ⎜ ⎟ ⎝ αA ⎠ ⎝ α̲ B ⎠

(4)

where [M]f is the free ion concentration of species M, and m and x are the valence of cation M and X. As an example, consider the a binary system of K+−Ca2+involving the exchange of K+ and Ca2+ in the presence of Cl−. [K]t = [K+]f (1 + KSKCl[Cl−]f )

(5)

[Ca]t = [Ca +]f (1 + KSCaCl[Cl−]f )

(6)

−

[Cl−]f =

[Cl ]t 2+

1 + [Ca ]f K sCaCl+ + [K+]f K sKCl

(7)

32

De Robertis et al. calculated the formation constants of [NaCl], [KCl]°, [MgCl]+ and [CaCl]+ ion pairs from different techniques at different temperatures and ionic strengths. All the values of αM (degree of dissociation) were fitted for each metal ion by the following equation: αM = 1 − [a1 + b1(T − 25)]C1/2 − [a 2 + b2(T − 25)]C + a3C 3/2

(8)

The parameters for eq 8 defining the dependence of αM on total concentration of NaCl, KCl, MgCl2 and CaCl2 are tabulated in Table 1. Table 1. Parameters for eq 8 Defining the Dependence of α on Total Concentration of NaCl, KCl, MgCl2 and CaCl2 M +

Na K+ Mg2+ Ca2+

a1

a2

a3

b1

b2

0.0330 0.0270 0.4160 0.2580

0.2190 0.2460 0.8120 0.7650

0.0790 0.0590 0.6280 0.2300

−0.0009 −0.0005 +0.0039 +0.0046

−0.0004 −0.0001 −0.0032 −0.0055

The stability constant is defined in terms of αM as

(1)

K sMX =

Here the underscore denotes the ion is in the exchanger phase. The equilibrium constant may be defined as T KAB

[MxX m] m+ x [M ]f [X x −]mf

KSMX =

2. ION EXCHANGE EQUILIBRIA MODELING Ion exchange between a solution and solid resin may be represented by the reversible stoichiometric equation of the type: z B A±̲ zA + zA B±z B ⇔ z B A±zA + zA B̲±z B

(3) 30,31

(1 − αM) (CMαM(αM + (z M − 1)))

(9)

The values of and αM stability constant and degree of dissociation at various temperatures are found by using eqs 8 and 9. Then the three equations, as mentioned above, eqs5, 6 and 7, may be solved to determine the concentrations of free ions available for the ion exchange for the K+−Ca2+binary system with Cl−. The activity in eq 2 for free ion Mm+ is related to the free ion concentration by KMX S

(2)

where α̲ A and α̲ B are the activities of ionic species A and B in the exchanger phase, respectively, αA and αB are the activities of species A and B in the solution phase, respectively, and zA and zB are valences of species A and B, respectively. B


Industrial & Engineering Chemistry Research αM

m+

= γMm+[Mm +]f

Article (1) (ϕ) interaction parameters (Aϕ, β(0) Ma, βMa and CMX). For computational ease, these functions may be recast into the following eight parameter expression using simple algebraic transformations:

(10)

Vo and Shallcross33 found that Pitzer model29 and Meissener and Kusik model28 were comparable for binary and ternary systems over ideal and Debye−Hückel.34 However, due to its incorporation of all interactions between ions in solution, the Pitzer model showed clear superiority over the Meissner and Kusik28 model in quaternary and five component systems. In the present study, the Pitzer model29 is used to calculate the activity coefficient of solution phase by incorporating the effect of temperature on Pitzer interaction parameters. The Pitzer model29 expression for the activity coefficient of the cation M in aqueous electrolyte mixtures involving c cation and a anion may be written as

Parameter (T ) = a1 + a 2T + a3/T + a4 ln T + a5/(T − 263) + a6T 2 + a 7(680 − T ) + a8/(T − 227)

The values of fitting constants in eq 19 for the binary (1) (ϕ) interaction parameters Aϕ, β(0) Ma, βMa and CMX as a function of temperature for aqueous electrolytes are listed in Table 2. Table 2. Values of the Fitting Constants eq 19 for the Binary Pitzer Interaction Parameters for Aqueous Electrolytes

Na

ln γM =

2 r zM f

+

∑ ma(2ΒMa + zCMa) a=1

+

Pitzer interaction parameters

∑ mc(2ΦMc + ∑ maψMca)+ c=1

a=1 Na

∑ ∑ a

Aϕ

Na

mama ′ψaa ′ M + |z M| ∑ ∑ mc maCca

a ′= a + 1

NaCl

β0

c=1 a=1

(11)

β01

The auxiliary terms may be calculated by Cϕ

⎛ ⎞ I 2 f = − Aφ ⎜ + ln(1 + b I )⎟ ⎝1 + b I ⎠ b r

+

∑ ∑

mc mc ′Φ′cc ′ +

c = 1 c ′= c + 1 Nc Na

+

∑ ∑

CaCl2

mama ′Φ′aa ′

c=1 a=1

Cϕ

(12)

MgCl2

where b is an empirical parameter equal to 1.2 at 25.0 °C. I is the ionic strength of solution and Aϕ may be defined as 35

I=

1 2

∑ mizi2 i

1.5 0.5 1 2πNAρw ⎛ e 2 ⎞ Aφ = ⎜ ⎟ 3 1000 ⎝ εkT ⎠

Β′Ma =

′ a I) g ′(aMa I ) g ′(aM ∂ΒMa = βM(1)a + βM(2)a I I ∂I

2(1 − (1 + x)e−x) , x2 −2(1 − (1 + x + (x 2/2))e−x g ′(x) = x2

Cϕ

(13) KCl

Cϕ

a1

a2

a3

a4

a5 0.3365 0.0022 14.3783 0.0000 −0.4830 0.0000 −0.1005 0.0000 −94.1895 −0.9228 3.4787 0.0000 19.3056 0.0882 0.5760 0.0000 2.6013 0.0000 0.0595 0.0000 26.7375 0.0000 −7.4155 0.0000 −3.3053 0.0000

a6 −0.0006 0.0000019 0.0056 −0.000002 0.0014 0.0000 −0.000018 0.000000034 −0.0404 0.000015 −0.0154 0.00003 0.0097 −0.000004 −0.00093 0.00000059 −0.0109 0.000026 −0.000249 0.00000024 0.0107 −0.00000375 0.0000 0.0000 −0.0012 0.00000049

a7 9.1425 45.2586 −422.1852 4.4385 119.3119 0.0000 8.6118 0.0683 2345.5036 −1.3908 0.0000 0.0000 −428.3837 9.9111 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 −758.4854 0.0000 322.8929 0.0000 91.2712 0.0000

a8 −0.0135 0.0000 −2.5122 −1.7050 0.0000 −4.2343 0.0124 0.2939 17.0912 0.0000 0.0000 0.0000 −3.5799 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 −4.7062 0.0000 1.1643 −5.9457 0.5864 0.0000

Mehablia et al.21 proposed the use of the Gaines and Thomas approach41 to independently calculate the equilibrium constant, which allows the decoupling of the equilibrium constant from the Wilson parameters. Then a two-parameter regression is performed to determine the Wilson interaction parameters. Later, Tsaur and Shallcross42 modified Mehabila’s approach by introducing the work of Argersinger and Davidson43 to replace Gaines and Thomas.41 The only difference is that Gaines and Thomas41 expressed the equation using equivalent ionic fractions whereas Argersinger and co-workers defined the equations in terms of mole fractions. The present study applies the method of Mehabila21 with a modification to evaluate the equilibrium constant and Wilson27 interaction parameters for each of the constitutive binary system. The Argersinger and Davidson43 approach is applied to calculate equilibrium quotient and is defined as

(15)

(16)

(17)

(φ) CMX

2 |z Mz X|1/2

β0 β1

(14)

g (x ) =

CMX =

β0 β1

where, NA is Avogardo’s number, ρw and ε are the density and the static dielectric constant of the pure solvent, respectively, k is Boltzmann’s constant, T is the absolute temperature and e is the electron charge. ′ a I) ΒMa = βM(0)a + βM(1)a g (αMa I ) + βM(2)a g (αM

β0 β1

a = 1 a ′= a + 1

∑ ∑ mcma Β′ca

(19)

(18)

36−40

Several workers have proposed empirical functions that describe the temperature variations of several of Pitzer’s C


Industrial & Engineering Chemistry Research ln KAB =

∫0

Article

1

ln λABdyB

equation from which the resin phase activity coefficient γi may be calculated if the parameters Λij are known. Equations 24 and 25 give the activity coefficients in a binary system as a function of the mole fractions in the exchanger phase and the Wilson interaction parameters: y1 ln γ1 = 1 − ln(y1 + y2 Λ12) − (y1 + y2 Λ12)

(20)

Here yB is the exchanger phase mole fraction of the species B and KAB and λAB are thermodynamic equilibrium constant and the equilibrium quotient, respectively. The equilibrium quotient is defined as λAB

z ⎛ y ⎞ zb ⎛ γ C B ⎞ a B A ⎜ ⎟ ⎜ ⎟ =⎜ ⎟ ⎜ ⎟ ⎝ γACA ⎠ ⎝ yB ⎠

−

(21)

where γA and γB are the activity coefficients of components A and B in solution phase, respectivelym CA and CB are the concentrations of components A and B in solution phasem respectivelym and yA and yB are mole fractions of components A and B in the exchanger phase, respectively. The equilibrium quotient is related to the thermodynamic equilibrium constant by the equation. KAB = λAB

−

y2 (y2 + y1Λ 21)

y1Λ12 (y2 Λ12 + y1)

(25)

Smith and Woodburn, along with other workers such as Shallcross et al.53 successfully used this formulation to determine the equilibrium constant and two binary interactions parameters via nonlinear least-squares regression algorithms. The value of the thermodynamic equilibrium constant and experimental data is used to determine the values of the Wilson binary interaction parameters Λ12 and Λ21 for a given system.

KAB may be obtained from the data of binary ion exchange equilibrium experiments and the logarithm of equilibrium quotient may be approximated by a taking a weighted mean of equilibrium quotient. 2.2. Exchanger Phase Nonidealities. Several methods have been proposed to model the nonideal behavior of the exchange phase. Bond44 proposed an empirical model based on the Rothmund−Kornfeld observation that the logarithm of the experimental equilibrium data for the solution and exchanger phases are linearly related. Novosad and Myers45 and then Myers and Byington46 considered the ion exchange system as essentially being an adsorption process with the exchanger consisting of two different types of adsorption sites. Later Melis et al.47 proposed that ion exchange media consists of two different types of exchange sites, each of which behaves ideally. Still later Lukey et al.48 applied a statiscial thermodynamics model originally proposed by de Kock49 to describe multicomponent sorption of gold cyanide and copper cyanide onto ion exchange media that is assumed to contain many different functional groups. Using an extensive set of experimental data for the binary, ternary and quaternary exchange between competing ions Vo50 compared the model predictions made by models using different methods to account for nonidealities in the exchanger phase. He found that the model developed by Wilson27 predictes equally as good as any other model. Elprince and Babcock51 first proposed applying the Wilson model27 to estimate the exchanger phase activity coefficients. The Wilson model was originally developed from statistical thermodynamics for the description of liquid−vapor equilibrium systems. The general form of the Wilson equation is m

(24)

52

(22)

⎡ ⎛ ⎞⎤ y1 y2 Λ12 T ⎟⎥ − z 2⎢1 − ln(y1 + y2 Λ12) − ⎜⎜ − ln K12 ⎢⎣ (y1Λ 21 + y2 ) ⎟⎠⎥⎦ ⎝ (y1 + y2 Λ12) ⎡ ⎛ ⎞⎤ y2 y1Λ12 ⎟⎥ + z1⎢1 − ln(y2 + y1Λ 21) − ⎜⎜ − ⎢⎣ (y2 Λ12 + y1) ⎟⎠⎥⎦ ⎝ (y2 + y1Λ 21) ⎡⎛ y ⎞ z 2 ⎛ C γ ⎞ z1⎤ 2 2 ⎟ ⎥ = ln⎢⎜⎜ 1 ⎟⎟ ⎜⎜ ⎢⎝ C1 γ ⎠ ⎝ y ⎟⎠ ⎥ 1 ⎣ ⎦ 2

(26)

The Wilson binary interaction parameters are determined for each binary system by optimizing eq 27 against the objective function:

OFw =

2 N M ⎛ ye − yp ⎞ ∑i = 1 ∑ j =i 1 ⎜ y ⎟ wj ⎝ e ⎠j

NM − 1

(27)

Here OFw is the weighted objective function, N is the number of species existing in the system and M is the number of experimental data. wj is the weighting function, which is defined below. ye and yp are the experimentally determined and the predicted composition of the exchanger phase. The present work uses a weighting function based on the Euler equation:33 w = 1 − e−ky(1 − y)x(1 − x)

(28)

The above weighting function favors the more reliable central data at the expense of the less reliable data at the extreme ends of the composition range. The use of this Euler function also helps flexibly vary the range of the reliable data by changing the k-value. The value of k = 25.0 is used. The primary advantage of the Wilson equation is that Wilson interaction parameters, Λij and Λji, are estimated from binary experimental data and may be applied directly to calculate the activity coefficient of multicomponent systems. 2.3. Incorporation the Effect of Temperature on Ion Exchange Equilibria. A model is developed by studying the effect of temperature on the equilibrium constant and

m

ΔGE = −∑ [ymi ln(∑ Λijymi )] RT i=1 j=1

(y1Λ 21 + y2 )

ln γ2 = 1 − ln(y2 + y1Λ 21) −

γAzb γBza

y2 Λ12

(23)

where GE is the molar excess Gibbs free energy of mixing, yi is the mole fraction of component i, m is the number of the components in the system and the Λij is the Wilson interaction parameter. Differentiation of the above equation with respect to the number of moles of species i in the resin phase yield the D



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equilibrated solution was determined by using an ICP (inductively coupled plasma) technique. Several experiments were conducted using the same amount of resin, 0.0500 g, with 100 mL of magnesium chloride solution of increasing concentrations. This process was undertaken in triplicate to verify the reproducibility of results. The cation exchange capacity was found to be 4.8 mequiv/dry g of resin with the relative standard error of 1.5%. This process is undertaken in triplicate to verify the reproducibility of results. Binary ion exchange equilibrium experiments were conducted for binary combinations of the Na−K−Mg−Ca systems, i.e., Na−K, K−Mg, K−Ca, Na−Ca, Na−Mg, Ca−Mg by using batch technique. The air-dried resin was weighed ranging from 0.05 to 70 g and put into 15 to 18 various flasks and precisely 100 mL of the prepared electrolyte solution were added to each flask using volumetric pipettes. The flasks were sealed by membranes (paraffin at 4 °C and aluminum foil at 60 °C) and shaken continuously in a temperature-controlled shaker at fixed rpm for at least 72 h to allow the ions in the solution and exchanger phases to come to equilibrium. These experiments were conducted at a concentration of chloride solution of 0.1 N at temperatures of 4, 25 and 60 °C. The equilibrium composition of the cations on the exchanger phase was calculated by a simple material balance. The lower temperature boundry of 4 °C was set, as it is just above the freezing point of the solution whereas the upper temperature boundry of 60 °C was imposed by the properties of the exchange resin used. Above this temperature, the physical properties of the material begin to change significantly. A single concentration of 0.1 N was selected, as previous studies by the authors33,42 have found that concentration to be worthy of further study. Also, the values required for certain model parameters such as the dissociation constant and Pitzer interaction parameters are available.

interaction parameters and associating it with thermodynamic functions. The Gibbs free energy is defined as 0 ΔGAB = −ln KAB RT

(29)

This relationship implies that ln KAB varies linearly with the inverse of absolute temperature.

⎡ b ⎤ ln ΛAB = ⎢aAB + AB ⎥ ⎣ T ⎦

(30)

⎡ b ⎤ ln ΛBA = ⎢aBA + BA ⎥ ⎣ T ⎦

(31)

Equations 30 and 31 represent the modified expressions for the Wilson parameters for a binary system by assuming a linear relation between the Wilson binary interaction parameters and temperature. In the present study, the natural logarithm of these parameters are plotted against the inverse of absolute temperature for each binary system in order to develop the relationship between thermodynamic equilibrium constant and Wilson binary interaction parameters with respect to temperatures ranging from 4 to 60 °C. 2.4. Evaluation of Error. A weighted relative residue is used to evaluate and quantify the accuracies of the predictions of the ion exchange equilibrium by using proposed temperature-dependent and experimental data. Weighted relative residue is defined as

RR =

2 ⎤ ⎡ M N ⎛ χmodel − χexpt ⎞ ⎟ w⎥ ∑i = 1 ⎢∑i = 1 ⎜ χ ⎝ ⎠ j j ⎥⎦ expt ⎢⎣ i

MN − 1

(32)

4. RESULTS AND DISCUSSION The equilibrium behaviors of six binary systems Na+−K+, K+− Ca2+, K+−Mg2+, Na+−Mg2+, Na+−Ca2+, Ca2+−Mg2+ at 4.0, 25.0 and 60.0 °C with a total solution concentration of 0.10 N were studied experimentally. The experimental data was used to calculate the model parameters (thermodynamic equilibrium constant and Wilson binary interaction parameters) using the methodology as described in section 3. The model parameters are determined for six binary systems at three temperatures, 4.0, 25.0 and 60.0 °C, and are presented in Table 3. These model parameters are used to compare the experimental and theoretical fitted isotherms at three temperatures for all six binary systems. Due to the limitation of space, three systems are presented here, which represent the 1:1 and 1:2 systems in Figures 1, 2 and 3. These figures show good agreement between model fitting and the experimental observation for each binary system. Figure 1 represents the system of K−Na and found this system has no effect of temperature on ion exchange equilibria because all isotherms overlap each other at all temperatures. It agrees with the theoretical calculations were made in present study to calculate the activity coefficients for cations by using Pitzer model and incorporating the effect of temperature on Pitzer interaction parameters. It is observed that for the system of K−Na, activity coefficients for both cations decrease as temperature increases for all concentrations and it cancels the effect of temperature on each other.

where M is the number of equilibrium data points, N is the number of cationic species and χ is quantity of interest. The weighted relative residues between experimental and predicted data is calculated by using eq 32 for all binary systems.

3. EXPERIMENTAL MATERIAL, APPARATUS AND PROCEDURE A simple batch technique was used to collect experimental data at three different temperatures for the six binary systems involving Na+, K+, Ca2+ and Mg2+ ions with Cl− as the nonexchanging ion. In the present study, Amberjet 1200H (geltype) was used as the ion exchange medium due to its stable characteristics and purchased from Rohm and Hass. Before starting the equilibrium experiment, the resin is preconditioned and converted into the desired form by using the column technique to remove any water-soluble residues or undesired cations remaining on the resin after the manufacturing process. All solutions were prepared by using analytical grade chemicals including crystallized salts such as sodium chloride, magnesium chloride and calcium chloride and ultrapure water. The moisture content of the resin samples was determined. To determine the cationic exchange capacity, the air-dried resin in a known cationic form (Na+) was weighed and put in contact with 100 mL of known concentration of solution in the Erlenmeyer flask. Sealed flasks containing the solution and resin beads were shaken without interruption in a temperature controlled shaker for 72 h at fixed rpm in order to allow the system to equilibrate. The cation concentrations in the E



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Table 3. Thermodynamic Equilibrium Constant and Wilson Binary Interaction Parameters for All Six Binary Systems model parameters system Mg2+−Na+

Ca2+−Mg2+

Ca2+−Na+

Ca2+−K+

Mg2+−K+

K+−Na+

temperature (°C)

KAB

ΛAB

ΛBA

no. data point

4.0 25.0 60.0 4.0 25.0 60.0 4.0 25.0 60.0 4.0 25.0 60.0 4.0 25.0 60.0 4.0 25.0 60.0

1.99 3.56 5.93 1.37 1.51 1.56 7.04 13.47 19.23 2.56 4.35 6.89 1.55 1.58 1.86 1.23 1.25 1.30

1.09 0.66 0.67 1.15 0.83 0.81 0.72 0.64 0.96 0.77 0.61 0.47 1.84 1.56 1.45 1.30 1.31 1.19

1.26 1.04 1.50 0.46 0.91 0.99 0.97 1.17 1.26 1.44 1.61 1.73 0.25 0.45 0.57 1.08 0.93 0.99

18 17 19 17 18 25 13 12 11 11 12 12 16 11 18 9 11 12

Figure 2. Comparison of experimental and fitted isotherms for the system of Ca−K at solution concentration of 0.10 N and various temperatures.

Figure 3. Comparison of experimental and fitted isotherms for the system of Mg−K at solution concentration of 0.10 N and various temperatures.

Figure 1. Comparison of experimental and fitted isothermsfor the system of K−Na at solution concentration of 0.10 N and various temperatures.

temperature has no significant effect and this can easily seen by the gradients of the curve. Figure 5 represents a relationship between thermodynamic equilibrium constant, Wilson binary interaction parameters with respect to temperature for the Ca− K system. It can be seen by the gradients of the curve that a linear equation can be used to describe the variation in the regression parameters with changes in the inverse of absolute temperature. These expressions may be used to describe the model parameters at the temperature between the extreme temperatures. Figure 6 represents a relationship between thermodynamic equilibrium constant, Wilson binary interaction parameters with respect to temperature of the Mg−K system with respect to temperature at a solution concentration of 0.10

Figure 2 represents the Ca−K system and it is found that this system has decreased selectivity of resin at low temperatures and increased selectivity at high temperatures. Figure 3 represents the K−Mg system and it is observed that this system has decreased selectivity of resin at low temperatures and increased selectivity at high temperatures. The natural logarithm of these parameters are plotted against the inverse of absolute temperature for each binary system in order to develop the relationship between thermodynamic equilibrium constant, Wilson binary interaction parameters with respect to temperature. These plots are presented in Figures 4, 5 and 6. Figure 4 represents the Na−K system (1:1) and it is concluded for the systems of same valences that F



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Figure 4. Model development for the system of K−Na with respect to temperature at a solution concentration of 0.10 N.

Figure 5. Model development for the system of Ca−K with respect to temperature at a solution concentration of 0.10 N.

interaction parameters. In the present study, we find that the effect of temperature is most pronounced for exchange between ions of different charges. For example, as the temperature increases from 4 to 60 °C, the value for the thermodynamic equilibrium constant for exchange between K+ and Na+ ions increases by just 5% whereas for exchange between Ca+ and K+ ions, the increase is over 165%. The temperature dependence of the equilibrium constant may be calculated from the

N. The linear fitting is done, and the equations determined for all the systems are presented in Table 4. These equations demonstrate the relationship and variation of thermodynamic equilibrium constant and Wilson binary interaction parameters with respect to temperature. The shapes of the binary isotherms presented in Figures 1−3 are dictated primarily by the value of the equilibrium constant, KTAB, and, to a lesser extent, the values of the Wilson binary G



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Figure 6. Model development for the system of Mg−K with respect to temperature at a solution concentration of 0.10 N.

Table 4. Summary of the Relations for Equilibrium Constant and Wilson Interaction Parameters with Respect to Temperature system (A − B) K−Na Ca−K Ca−Mg Mg−Na Ca−Na Mg−K

ln ΛAB

ln ΛBA

−0.275153.662/T 1.447296.39/T −1.937 + 556.9/T −2.734 + 749.844/T 1.443 − 516.3/T −0.790 + 381.2/T

−0.412 + 123.509/T −3.192 + 809.33/T 3.712 − 1206/T 1.335 − 334.231/T 1.506 − 417.5/T 3.455 − 1316/T

ln KAB 0.538 6.793 1.076 7.152 7.851 1.537

− − − − − −

92.508/T 1609/T 206.5/T 1778/T 1610/T 310.7/T

enthalpy of the ion exchange reaction via the well-known van’t Hoff equation. As Agrawal and Sahu2 note, the change of enthalpy associated with the ion exchange process may arise from a number of sources: • the difference in heat consumed in breaking the bonds of different ions as they are released from the exchanger phase; • the difference in heat released in the formation of the bonds as the different ions become attached to the exchange phase sites; • the heat associated with any change in the interlayer distance; • the enthalpy changes associated with hydration or dehydration in the solution phase. The results of the present study suggest that the process of displacing K+ ions on the exchanger phase with Ca2+ ions is endothermic, i.e., the preference the exchanger phase displays for the Ca2+ ions over the K+ ions increases with increasing temperature (Figure 2). Similar observations may be made about the other binary systems. The equations are used to produce the predictions of the ion exchange curves for each system at other temperatures within the range of 4 to 60 °C. The predicted values derived from the proposed model for all binary system at all temperatures are presented in Table 5.

Table 5. Predicted Values of Thermodynamic Equilibrium Constant and Wilson Binary Interaction Parameters for All Six Binary System at a Solution Concentration of 0.10N model parameters system

temperature (°C)

KAB

ΛAB

ΛBA

Mg2+−Na+

4.0 25.0 60.0 4.0 25.0 60.0 4.0 25.0 60.0 4.0 25.0 60.0 4.0 25.0 60.0 4.0 25.0 60.0

2.08 3.27 6.12 1.39 1.47 1.57 7.70 11.59 20.45 2.66 4.02 7.08 1.52 1.64 1.83 1.23 1.26 1.30

0.97 0.80 0.62 1.07 0.93 0.76 0.65 0.75 0.89 0.76 0.62 0.46 1.79 1.63 1.42 1.32 1.27 1.21

1.13 1.24 1.39 0.50 0.72 1.09 1.00 1.11 1.29 1.45 1.56 1.73 0.27 0.38 0.61 1.04 1.00 0.96

Ca2+−Mg2+

Ca2+−Na+

Ca2+−K+

Mg2+−K+

K+−Na+

H



Article

First, these predicted values of the model parameters are used to predict the ion exchange equilibria for all binary systems at experimental temperatures to verify the accuracy of the model. Figure 7 represents the comparison of experimental

Figure 8. Predicted isotherms for the system of Ca−K at various temperatures and at a solution concentration of 0.10 N.

Figure 7. Comparison of experimental and predicted isotherms for the system of Ca−K at a solution concentration of 0.10 N and temperature of 4.0 °C.

and predicted isotherms for a system of Ca−K at 4.0 °C. A weighted relative residue of 0.010 53 is found and it lies within the experimental error. It is concluded from the above observation that the accurate prediction of ion exchange equilibria of binary system of Ca−K can be made. Second, by using this model, the binary system can be predicted at other temperatures as well. Figure 8 represents the prediction of different isotherms at various temperatures for the system of Ca−K by using the predicted model parameters derived from the proposed model. It is concluded from the observation that the selectivity of resin is increased by increasing the temperature. Figure 9 represents the prediction of different isotherms at various temperatures for the system of Mg−Na. The variation in the equilibrium isotherm shape with increasing temperature for the binary systems studied generally reflects the trends seen in the shapes of the isotherms when solution concentration is decreased. Working with different cation resins, several workers 33,42,53 have found that as the total solution concentration decreases, the preference exhibited by the exchanger for the multivalent ion over the monovalent ion increases. Thus, the increase in preference for the multivalent ion over the monovalent ion as temperature is increased from 4 to 60 °C is similar to the increase in preference as the solution concentration is decreased from 0.1 to 0.01 N. Figure 10 presents the relationship between the natural logarithm of product of Wilson interaction parameters and inverse of absolute temperature. For the binary systems of Ca− K, K−Na and Ca−Mg, the natural logarithm of the product of Wilson binary interaction parameters have a linear relationship with respect to the inverse of absolute temperature. Figure 10 explains that no linear relationship is found for other binary

Figure 9. Predicted isotherms for the system of Mg−Na at various temperatures and at a solution concentration of 0.10 N.

systems. According to the Hála constraint, the natural logarithm of the product of the Wilson binary interaction parameters should be equal to zero. Figure 10 shows that the natural logarithm of the product of the Wilson binary interaction parameters is not constant.

5. CONCLUSIONS The temperature-dependent behavior of six binary systems Na+−K+,K+−Ca2+, K+−Mg2+,Na+−Mg2+, Na+−Ca2+ and Ca2+− Mg2+ was investigated using the Amberjet 1200H resin. It is found that temperature has no significant effect on 1:1 valences systems such as Na+−K+ and Ca2+−Mg2+. It is observed that temperature has a significant effect for the systems of 1:2. A I



Article

Figure 10. Relationship between Hála constraint and temperature.

m = number of components NA = Avogadro’s number R = universal gas constant T = absolute temperature, K wj = weighting factor X− = univalent anion xi = equivalent ionic fraction of species i in solution phase yi = equivalent ionic fraction of species i in exchanger phase ymi = exchanger phase mole fraction of species i zi, zM = valences of component i or M

semiempirical thermodynamic model incorporating temperature dependence is proposed. It is found that proposed model is in good agreement with the experimental data for all systems and temperature investigated by using the relative residue technique. The relative residue lies within the experimental error range of 0.006 to 0.0215. The selectivity of resin is increased by increasing temperature for all cations, and the order of preference for the cations is Ca 2 + > Mg 2 + > K+ > Na +

The concepts behind the model proposed are equally applicable to the exchange of both anions and cations, either individually or in mixed beds.

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Greek

αi = activity of component i in the solution phase αi = activity of component i in the exchanger phase αM = degree of dissociation (1) (2) β(0) Ma, βMa, βMa = Pitzer correlation parameter σ = variable power of the weighting function γi = activity coefficient of component, i in the solution phase γi = activity coefficient of component i in the exchanger phase ε = static dielectric constant E θij(I), Eθij(I) ′ = Pitzer correlation parameter λAB = equilibrium quotient Λij = Wilson interaction parameter ρw = density of the solvent Φ = Pitzer model parameter Ψ = Pitzer model parameter χ = quantity of interest

AUTHOR INFORMATION

Corresponding Author

*D. C. Shallcross. E-mail: [email protected]. Telephone: +61 3 8344 6614. Fax: +61 3 8344 4153. Notes

The authors declare no competing financial interest.

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NOTATION Aφ, BMa, B′Ma, Cca, Cφca = Pitzer correlation parameter CM = molar concentration of M cation G = Gibbs free energy I = ionic strength e = electronic charge T KAB = thermodynamic equilibrium constant for binary system A−B KMX S = stability constant k = Boltzmann’s constant M+ = univalent cation MX = ionic compound

Subscript

a, a′, a″ = anion c, c′, c″, M = cation f = free ion J



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Article

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Temperature-Dependent Model for the Prediction of Binary Ion

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