Determination of Activity Coefficients, Osmotic Coefficients, and

Article pubs.acs.org/jced

Cite This: J. Chem. Eng. Data XXXX, XXX, XXX−XXX

Determination of Activity Coefficients, Osmotic Coefficients, and Excess Gibbs Free Energies of KCl and KBr Aqueous Mixtures at 298.15 K Rahman Salamat-Ahangari,*,† Laya Saemi Pestebaglu,† and Zohreh Karimzadeh*,‡ †

Department of Chemistry, Faculty of basic Sciences, Azarbaijan Shahid Madani University 35 km Tabriz-Maraqe Road, P.O. Box 53714161, Tabriz, Iran ‡ Department of Chemistry, Shahid Beheshti University, G.C., Evin-Tehran, Iran ABSTRACT: Isopiestic vapor pressure measurements are performed to study the ternary KCl + KBr + H2O system at 298.15 K. The modeling purposes are achieved based on the experimental osmotic coefficients regression of the ternary system. The present calculations are conducted for electrolyte molalities ranging from 0.3 up to about 4.9 mol kg−1 with various ionic strength fractions of KCl (yB = mKCl/(mKCl + mKBr) = 0.25, 0.51, and 0.75) . Finally, mean activity coefficients (γ±), osmotic coefficients (φ), and excess Gibbs free energies (Gex) are calculated and compared using the extended Pitzer (EP) and the Pitzer−Simonson−Clegg (PSC) equations.

1. INTRODUCTION

precised isopiestic measurement for a wide range of molalities and various ionic strength fractions.3−5 In this work isopiestic vapor pressure measurements performed on the ternary KBr + KCl + H2O system at 298.15 K with KCl(aq) served as reference standard solution, and total molalities ranging from 0.3 up to 4.95 mol kg−1 in a series of mixed electrolyte system characterized by its ionic strength fraction of KCl (yB = 0.25, 0.51 and 0.75). As a first step, the experimental data are correlated by use of an extended form of the Pitzer model (Archer form). The mole fraction-based model of Pitzer−Simonson−Clegg has also been chosen for data analysis.

Thermodynamic properties (particularly activity and osmotic coefficients) of aqueous mixed electrolyte solutions are needed to explore the nature of various ionic interactions in many industrial and environmental processes. The desalination, chemical separation, marine chemistry, geology, and environmental aerosol sciences are some of them.1 In view of the usefulness of these systems, it seems particularly interesting to explore mixed electrolyte solutions. Our group has recently reported data for the activity and osmotic coefficients of the aqueous mixtures of NaCl and NaBr in aqueous electrolyte solutions.2 In this work, isopiestic vapor pressure measurements are performed on the ternary KBr + KCl + H2O system at 298.15 K A review of literature shows that isopiestic vapor pressure measurements were made at 298.15 K for solutions of the salt pairs KBr + KCl, by Covington et al.4 McCoy and Wallace measured activity coefficients of KCl and KBr in the KCl + KBr + H2O system using the isopiestic method at total molalities extending from 2 mol kg−1 up to saturation.3 The data of Covington et al. cover a narrow range of molalities and the results are somewhat scattered. The McCoy and Wallace results were focused on Harned’s rule using approximated constant total molalities, in addition their data were significantly scattered. The published results of these two investigators are not in agreement (osmotic coefficients of these two studies differ by 2−3% in the overlap region); one of the main source of these inconsistency seems to be due to low-purity chemicals. Consequently we were prompted to extend and supplement the © XXXX American Chemical Society

2. EXPERIMENTAL PROCEDURE The materials in this work are described in Table 1, and they were used without further purification. These salts are dried overnight in an oven at 110 °C, and double distilled deionized water is used as solvent. The isopiestic apparatus in this Table 1. Purities and Sources of the Chemical Samples Used in This Work chemical

source

purity (mass %)

KCl KBr NaCl

Sigma Merck Merck

99.99 99.99 99.99

Received: May 21, 2017 Accepted: December 4, 2017

A

DOI: 10.1021/acs.jced.7b00460 J. Chem. Eng. Data XXXX, XXX, XXX−XXX

Journal of Chemical & Engineering Data

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research is a laboratory built chamber based on those described elsewhere.5,6 The chamber is constructed from a glass desiccator and a gold-plated copper heat-transfer block (19 cm diameter and 2 cm thick) into which sample cups are inserted. Sample cups are made of a gold-plated silver cup (12 dishes) and are suitably accommodated in recesses drilled into the block. Each dish in which equilibrations are made has a diameter of 26 mm and a depth of 18 mm. Two pairs of dishes contained pure KCl (referred to as referenced standard solution), two pairs of dishes contained pure NaCl, and the remaining dishes contained mixtures of KCl and KBr with given molar ratios.To start a run, sufficient pure water is added to each dish to produce initial solutions approximately at the desired isopiestic concentrations. To prevent concentration gradient, a glass ball is placed in each dish, and also the air above the dishes is stirred. The chamber is evacuated and then slowly rocked back and forth in a water bath kept at 298.15 ± 0.01 K using the thermostated system (Julabo M12 Germany). Additionally, to prevent splattering the solution during evacuation, two hollow vessels are devised between the chamber and pump. After equilibration (lasting about 2 to 20 days) the vacuum is broken, and Teflon lids are placed on the dishes. This assembly is then weighed by an electronic balance (Sartorius ±0.0001 precision). For all cases buoyancy correction is made, and the reported molalities are the average of duplicate sample electrolytes. Each isopiestic molality is known at better than ±0.01%.

ϕ − 1 = {2/(mc + ma + ma ′)}[−Aϕ I3/2 /(1 + bI1/2) Tϕ Tϕ ϕ ϕ + mcma (Bc,a + ZCc,a ) + mcma ′(Bc,a ′ + ZCc,a ′) ϕ + ma ma ′(ϕa,a + mc Ψc,a,a ′)] ′

T T ln γc = zc2F + ma (2Bc,a + ZCc,a ) + ma ′(2Bc, a ′ + ZCc,a ′) T T + ma ma ′Ψc,a,a ′ + zcmcma Cc,a + zcmcma ′Cc,a ′

ϕ − 1 = −|z Mz X|Aϕ I +m

2

1/2

T lnγa = za2F + mc(2Bc,a + ZCc,a ) + ma ′(2ϕa,a ′ + mc Ψc,a,a ′) T T + |za|mcma Cc,a + |za|mcma ′Cc,a ′

/(1 + bI ) +

where F = −Aϕ{I1/2 /(1 + bI1/2)(2/b) ln(1 + bI1/2)} T′ T′ + mcma (B′c,a + ZCc,a /2) + mcma ′(B′c, a ′ + ZCc,a ′/2)

(8)

and the functions appearing in the eqs 6−8 are defined by (1) ° + βMX BMX = βMX gMX (α I1/2)

g (x) = (2/x)(1 − (1 + x)exp(−x)) (1) B′Mx = (βMX /I)g ′(x)

T (1) ° + 4CMX CMX = CMX h(ωI1/2)

(12)

and the function h(x) as given in the Appendix I of Clegg et al.;9 h(x) = (1/x 4)[6 − {6 + x(6 + 3x + x 2)} exp( −x)]

(1) 4 ° + 4{CM,X + m2(2νM2νX z M /ν){3CM,X /(ωM,X I2)}[6 − (6 + 6ωM,X I1/2

(13)

(2)

and

where M denotes K+ and X denotes Cl− or Br−; I is the ionic strength; b = 1.2 kg1/2·mol−1/2, Aϕ = 0.391475 kg1/2·mol−1/2 is the Debye−Huckel limiting-law slope for osmotic coefficients at 298.15 K, ZM and ZX are the valences of ions M and X, vM and vX denote the number of M and X ions formed by stoichiometric dissociation of one molecule of MX; and v = vM + vX .

° + = CMX

(11b)

and

2 [1 − (1 + αM,X I1/2 − αM,X I/2)exp( − αM,X I1/2)]}

T′ (1) CMX = (4/I)CMX h′(x)

(14a)

where h′(x) = exp( −x)/2 − 2h(x)

(14b)

Consequentially, the activity coefficients of MX (KBr or KCl) salts, may be calculated by combining individual ionic activity coefficients according to

and

(1) CMX exp( −ωMX I1/2)

(11a)

g ′(x) = exp( −x) − g (x)

(1) 2 ° + (2βMX + m(2νMνX /ν){2βM,X /αM,X I)

Tϕ CMX

(10)

where

lnγ± = −|z Mz X|Aϕ{I1/2 /(1 + bI1/2)(2/b) ln(1 + bI1/2)}

(1) ϕ ° + βMX BMX = βMX exp( −αMX I1/2)

(9)

where MX stands for ca or ca′ and g(x) function;

(1)

2 3 4 + 3ωM,X I + ωM,X I3/2 − ωM,X I2 /2) exp(− ωM,X I1/2)]}

(6)

T lnγa ′ = za2′F + mc(2Bc,a + ZCc,a ′) + ma (2ϕa,a ′ + mc Ψc,a,a ′) ′ T + |za ′|mcma Cc,a + |za ′|mcma ′CcT,a ′ (7)

ϕ m(2νMνX /ν)BMX

Tϕ (4νM2νXz M /ν)CMX

(5)

Analogous expressions for bromide and chloride ions are as follow:

3. MODELING 3.1. Extended Pitzer Model (EP). The Archer extension of the Pitzer ion-interaction model which includes the ionic strength-dependent third virial coefficients7 may be written in the following forms for the osmotic coefficients and activity coefficients of binary MX + H2O electrolyte solution: 1/2

(4)

γ±MX = (γMγX)1/2

(3)

(15)

The extended Pitzer (EP) model parameters for pure aqueous (1) (2) (0) (1) salt solutions are β(0) MX, βMX, βMX, CMX, and CMX. Also the two and three particles interaction parameters, mixture parameters, namely, ϕa,a′ = θBr,Cl and ψc,a,a′ = ψK,Br,Cl are determined by least-squares fit of experimental data (osmotic coefficient).

where αMX = 2 kg1/2·mol−1/2 and ωMX = 2.5 kg1/2·mol−1/2 for all cases at 298.15 K. Moreover, generalized equations for the extended form of Pitzer’s model for electrolyte mixtures are given in the paper of Clegg et al.9 as B



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3.2. Pitzer−Simonson−Clegg (PSC). A newer model for mixtures of symmetrical charge type electrolyte was developed by Pitzer and Simonson.10,11 This model was applicable over the entire concentration range. Their approach consists of a Debye−Hückel (DH) expression as a long-range force term and a three-suffix Margules expansion due to short-range forces. To include arbitrary charge type electrolytes, Clegg and Pitzer have generalized their model containing an indefinite number of ionic and neutral species.12,13 In this model, known as Pitzer−Simonson−Clegg (PSC), different long-range and short-range terms are used. An extended DH expression for long-range forces, as well as a four-suffix Margules expansion for short-range forces are combined which made their approach generally work very well for pure and mixed aqueous electrolytes and are as follows:12 An extended DH expression including composition-dependent terms is the contribution of long-range forces. A four-suffix Margules expansion which includes parameters for the interactions of both the solvent-anion and solvent-cation is used for the contribution of short-range forces. The dissociation of all electrolytes is considered completely and the following relation is used to convert the mole fraction based activity coefficient i to the molal based activity coefficient.

LR f MX = exp(−Ax(2/ρ log(1 + ρIx·5) + Ix1/2(1 − 2Ix)

/(1 + ρIx1/2)) + .5(x X + xM)BMX g − .5(x XBMX + x YBMY )(g + (2Ix − 1)exp(−z))) (20a) SR f MX = exp(X1((1 − FXx I)W1MX − x IFYW1MY )

+ XIx1(1 + FX − 2FXx I)U1MX + XIx1FY(1 − 2XI)U1MY + XIx12(1 + FX(1 − 3XI))V1MX + XIx12FY(1 − 3XI)V1MY + .5XIFY(1 − XIFX)WXYM + .5FYx I 2(2(FX − FY )(1 − x IFX) + FY )UXY + XIx1FY(1 − 2XIFX)Q 1XYM − W1MX)

An analogous expression for KBr is as follow: SR LR ln fMY = ln f MY + ln f MY

LR f MY = exp(−Ax (2/ρ log(1 + ρIx1/2) + Ix1/2(1 − 2Ix)

/(1 + ρIx·5)) + .5(x Y + xM)BMY g − .5(x XBMX + x YBMY )(g + (2Ix − 1) exp(−z))) SR f MY = exp(x1((1 − FYXI)W1MY − XIFXW1MX)

(16)

+ XIx1(1 + FY − 2FYXI)U1MY

where γi, MS, and mi are the molal activity coefficient, the mean molar mass of the solvent (water), and the molal concentration of the ionic solute species i, respectively. g E = g S + g DH

+ XIx1FX (1 − 2XI)U1MX + XIx12(1 + FY(1 − 3XI))V1MY + XIx12FX(1 − 3XI)V1MX + .5XIFX(1 − XIFY )WXYM − .5FXXI 2(2(FY − FX )(1 − XIFY ) + FX)UXY

(17)

+ XIx1FX(1 − 2XIFY )Q 1XYM − W1MY )

⎛ ⎞ ⎞ 1 ⎜ ⎛ ln f i* = ∂⎜⎜∑ nig E⎟⎟ /∂ni⎟⎟ ⎜ RT ⎝ ⎝ i ⎠ ⎠

(18)

Along with eqs 17, 18, and 19 the expression for the activity coefficient of water in solution KCl + KBr + water is given by ln f 1 = ln f DH + ln f SR f

= exp(2Ax Ix

/(1 + ρIx

1/2

(22)

g = 2(1 − (1 + z) exp(−z))/(z 2)

(23)

XI = 1 − x1

(24)

The mole fraction ionic strength (I) is equal to 1/2∑ixiz2i , and anion fractions FY and FX are defined for fully symmetrical systems as

) − xMx XBMX exp(−z) (19a)

f SR = exp(XI(1 − x1)(FXW1MX + FYW1MY )

Fx = x X /(x X + x Y )

(25)

FY = 1 − Fx

(26)

For the ternary system, with pure H2O as solvent, the used value of the parameter related to the ‘‘closest approach” distance is ρ = 14.0405, the standard constant value used for the parameter α is α = 13, and the Debye−Hückel parameter on the mole-fraction basis Ax = 2.9165 for water at 25 °C. BMX is the long-rang parameter of electrolyte MX in a very dilute range. In fact, W1MX, V1MX, and U1MX parameters describe short-rang interactions between MX and water. These four parameters are determined by fitting the experimental osmotic coefficient data of the binary system for salt MX in water. The mixture parameters for MX−MY−solvent are UXY, WXYM, Q1XYM., describe the short-rang interactions between MX, MY, and water. The mixture parameters are determined by

+ XI 2(1 − 2x1)(FXU1MX + FYU1MY ) + x1XI 2(2 − 3x1)(FXV1MX + FYV1MY ) − .5XI 2FXFYWXYM − XI 3FXFY(FX − FY )UXY + XI 2(1 − 2x1)FXFYQ 1XYM)

z = αIx .5

The total mole fraction of ions (xI) is given by

(19)

− xMx YBMY exp(−z))

(19b)

where MX stands for KCl, MY stands for KBr, and 1 stands for water. The corresponding expression for ionic activity coefficients of the KCl is LR SR ln fMX = ln f MX + ln f MX

(21b)

and the functions appearing in the eqs 19−21 are defined by

T ,P

3/2

(21)

(21a)

⎛ ⎞ f i* = γi⎜⎜1 + (M s/1000) ∑ mi⎟⎟ ⎝ ⎠ i

DH

(20b)

(20) C



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Table 2. Isopiestic Total Molalities mtot of (KCl + KBr) Aqueous System for Different Ionic Strength Fractions of KCl (yB) and Osmotic Coefficients φ at T = 298.15 Ka mtota/mol·kg−1

mtot/mol·kg−1

(yB = 0.25)

φ

0.3204 0.4836 0.6064 0.8528 0.956 1.0028 1.5016 1.9892 2.5404 3.0544 3.5444 4.0084 4.522 4.8028 4.89

0.9096 0.9058 0.9045 0.9050 0.9070 0.9073 0.9159 0.9272 0.9415 0.9551 0.9702 0.9831 0.9985 1.0061 1.0090

a

(yB = 0.51)

mtot/mol·kg−1 φ

(yB = 0.75)

φ

0.9068 0.9043 0.9034 0.9021 0.9028 0.9040 0.9063 0.9118 0.9177 0.9322 0.9592 0.9786 0.9948 1.0053

0.4137 0.4695 0.6329 0.8635 0.9729 1.1463 1.7696 2.1119 2.5552 3.1477 3.7004 4.0377 4.6580 4.8295 4.9735

0.9048 0.9037 0.9012 0.8998 0.9002 0.9022 0.9131 0.9208 0.9320 0.9476 0.9630 0.9722 0.9902 0.9952 1.0008

b

0.3938 0.5071 0.6241 0.7465 0.9367 1.1031 1.2524 1.5194 1.8378 2.4281 3.4087 4.0781 4.5983 4.9513

a The standard uncertainties u are u(T) = 0.01 K, u(m) = 0.0008 mol·kg−1, and u(φ) = 0.001. bIn this fraction the result was discarded because one of the sample cups was tipped over.

fitting the experimental data. To this aim, first using the following equation and activity coefficients of water, osmotic coefficients are calculated as ln(aW ) = −(MW /1000)ϕ ∑ m i i

Table 3. Isopiestic Molalities and Osmotic Coefficients of Aqueous System of KCl(aq) as Reference Solutions Corresponding to Different Ionic Strength Fractions of KCl (yB) at Ta = 298.15 K ma0KCl

(27)

Second, the mixture parameters are determined by fitting the experimental osmotic coefficient data of ternary systems mixed of salt MX and MY in water. Finally, for modeling purposes, the related EP and PSC model equations are correlated by experimental data via an iteration minimization procedure for obtaining the model parameters using the following relation: ⎡ ⎤1/2 2 σ = ⎢(1/n) ∑ (ϕtheor − ϕexp) ⎥ ⎢⎣ ⎥⎦ n

(yB = 0.25)

(yB = 0.51)

0.3220 0.4866 0.6103 0.8591 0.9646 1.0118 1.5180 2.0140 2.5748 3.0955 3.5967 4.0661 4.5891 4.8712 4.9613

0.9051 0.9003 0.8987 0.8983 0.8989 0.8993 0.9060 0.9157 0.9289 0.9424 0.9561 0.9692 0.9839 0.9919 0.9945

0.3957 0.5096 0.6275 0.7498 0.9409 1.1078 1.2584 1.5289 1.8488 2.445 3.4358 4.1121 4.6421 4.9999

(28)

where n is the number of experimental data, used for the determination of the quality of regression (or the root of mean square deviation) of the fit.

m0KCl b

ϕKCl

(yB = 0.75)

ϕKCl

0.9024 0.8999 0.8986 0.8981 0.8987 0.9002 0.9020 0.9062 0.9122 0.9257 0.9516 0.9705 0.9854 0.9956

0.4151 0.4711 0.6348 0.8649 0.9743 1.1483 1.7743 2.1181 2.5643 3.1592 3.7146 4.0522 4.6759 4.8486 5

0.9018 0.9006 0.8985 0.8983 0.8990 0.9006 0.9107 0.9180 0.9287 0.9441 0.9593 0.9688 0.9864 0.9913 0.9956

a

The standard uncertainties u are u(T) = 0.01 K, u(m) = 0.0008 mol· kg−1, and u(φ) = 0.001 bThe content of sample cup was inadvertently spilt

4. RESULTS AND DISCUSSION The isopiestic vapor pressure measurement values are reported for ternary H2O(A) + KCl(B) + KBr(C) system at 298.15 K in Table 2. It is observed that the total molalities range from 0.3 up to 4.95 mol kg−1 in a series of mixed electrolyte systems. All the performed experiments are based on three different values of ionic strength fractions of KCl (yB): yB = mKCl /I

m0KCl ϕaKCl

In this work the isopiestic equilibration of binary KCl and NaCl and the ternary system under study was run contemporaneously, and as matter of fact the KCl binary solution could be termed a secondary reference standard. The NaCl binary solution plays the role of primary reference standard solution.1,2 The molal osmotic coefficients, ϕ, for H2O + KCl + KBr solutions are calculated by the isopiestic equilibrium equation: m0KCl φ= φ = RφKCl mKCl + mKBr KCl (30)

(29)

where I is the total ionic strength I = mKCL + mKBr. The reference standard solution is KCl(aq). In Table 3, the values of isopiestic molalities and osmotic coefficients for aqueous system of KCl(aq) as reference solutions in different ionic strength fractions are reported. The reported osmotic coefficient values are calculated by the ion-interaction model of Pitzer as extended by Archer7 and using ion-interaction parameters taken from Archer.8

where m0KCl and ϕKCl are equilibrium molality and osmotic coefficient of the potassium chloride binary solution (in this regard terms “reference” solution) and mKCl and mKBr are the D



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molalities of KCl and KBr in the ternary system. R denotes isopiestic ratio. Figure 1 shows the experimental osmotic coefficients of the mixtures and limiting binary solutions as a function of the total

Figure 2. Measured and calculated osmotic coefficients of the ternary (KCl + KBr)(aq) at total molality = 4 plotted versus ionic strength fraction of KCl: (∗) calculated without mixing parameters; (△) calculated using θBrCl = −0.0027 and ΨKBrCl = 0.001 values; (●) experimental data. Figure 1. Plot of experimental osmotic coefficients against ionic strength for the ternary system (KCl(aq) + KBr(aq) at T = 298.18 K. The experimental data for pure KBr was taken from ref 15.

ionic strength. It may be noticed that the trends are very similar to that of the ternary H2O + NaCl + NaBr system.2 However, it seems that the KCl osmotic coefficients are affected much more by the addition of KBr in comparison to the addition of NaBr on NaCl. The reported experimental results in this paper are treated with two approaches: first the ion-interaction model of Pitzerextended by Archer(EP),7 includes the ionic strength dependent third virial coefficients and second the Pitzer−Simonson− Clegg (PSC) model.12 To implement the EP model, first the single salt solution parameters are required. According to eq 15 the used (1) (2) (0) (1) parameters, β(0) MX, βMX, βMX, CMX, and CMX, for KCl and KBr are obtained using the literature data. Table 4 shows all the used parameters in this study. Finally using eq 4 the experimental data are fitted and the mixture parameters are evaluated. Following the methodology used by Rard and Miller16 and using eq 4 the osmotic coefficients of Covington et al.4 together with our experimental results are then fitted to obtain θBrCl = −0.00275 kg·mol−1 and ΨKBrCl = 0.001 kg2·mol−2 with σ = 0.0021. The influence of these mixing parameters is demonstrated in Figure 2 which shows measured and fitted osmotic coefficients at mT = 4 mol·kg−1, as a function of ionic strength fraction of KCl. It is evident from Figure 2 that the fitted model using the mixing parameters agrees well with the experimental points. Figure 3 represents a comparison of osmotic coefficients of our KCl as reference and experimental values of Robinson and Stokes and also the experimental data by Hammer and Wu. As can be seen, our results are in excellent agreement with

Figure 3. Plot of experimental osmotic coefficients of KCl+ H2O against ionic strength: (△) Robinson and Stokes;17 (□) Hammer and Wu;15 (◆) this work.

Hammer and Wu and marginally discrepant from Robinson and Stokes as quoted above. This might be attributed to their lower purity of the chemicals. The initial fits to evaluate mixing parameters of the Pitzer model are based on published isopiestic molalities of Covington et al.4 This fit is accomplished by using modern osmotic coefficient values of the potassium chloride standard as quoted above and using the ion-interaction parameters for potassium bromide which were critically assessed by Popovic et al.14 Covington et al. represent their measurements at high concentrations and their data points numbers are less than this study (narrow range from 2.03 up to 4). Correspondingly their redetermined osmotic coefficient values are included in Figure 4 and compared with those calculated by the extended Pitzer model of the present work, and they are fairly consistent (just three points are rejected in the least-squares fitting as

Table 4. Extended Pitzer (EP) Model Parameters for Aqueous KBr and KCl Solutions at 298.15 K Solute KCl KBr

β(0) MX

β(1) MX

β(2) MX

C(0) MX

C(1) MX

(kg·mol−1)

(kg·mol−1)

(kg·mol−1)

(kg2·mol−2)

(kg2·mol−2)

max molality

ref

0.0511 0.0564

0.2019 0.2110

0 0

−0.0007 −0.0010

0 0.0172

5.0 5.5

8 14

E



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Also using the extended Pitzer model, the mean activity EP coefficients for KCl and KBr (γEP ±KCl, γ±KBrl), osmotic coefficients EP (φ ) and excess Gibbs free energies (Gex,EP) are calculated and depicted in Table 5. The PSC model is the second approach to correlate the experimental data. As Clegg et al. discussed,13 the effects of quaternary coefficient, V1MX are limited to very concentrated systems. Our computations also confirm it. Therefore, we keep BMX, W1MX, and U1MX as the parameters which are determined by the binary solutions of the relevant salt in water using eq 19. Pitzer−Simonson−Clegg model parameters for aqueous KBr and KCl Solutions at 298.15 K are depicted in Table 6. KCl parameters are those used by Clegg et al.,13 but KBr parameters are computed using Hamer and Wu data.15 Table 6. Pitzer−Simonson−Clegg Model Parameters for Aqueous KBr and KCl Solutions at 298.15 K Figure 4. Differences between the experimental osmotic coefficients Φ for (yB KCl + yC KBr)(aq) and those calculated using eq 4 at T = 298.18 K, plotted against ionic strength. □, this work; (∗) Covington et al.;5 (+) McCoy and Wallace.4

outliers). On the contrary McCoy and Wallace did not represent their experimental results, and their data analysis was based on Harned’s rule. For comparison reasons the experimental values of the activity coefficient are calculated using their tabulated coefficients of Harned’s rule (α12 and α21).3 In addition we convert their relationship to an osmotic coefficients equation using the Gibbs−Duhem relation in order to calculate osmotic coefficients. The resulting data are represented in Figure 4. An inspection of Figure 4 reveals that the McCoy and Wallace data are high, and their results are quite discrepant from Covington et al. data, therefore their data points are assigned zero weight in the least-squares fit. As mentioned by Robinson the defect of the constant total molal measurements by the isopiestic method is its indirect evaluations (using graphical interpolation) of molalities, hence the procedure of McCoy and Wallace is significantly affected by this fact.17 Moreover the precision of the Covington group and McCoy Wallace isopiestic studies quoted above might be influenced by the low purity of the chemicals.

solute

BMX

W1MX

U1MX

max molality

ref

σ

KCl KBr

9.991 4.921

−3.269 −2.946

−2.283 −1.673

5.0 5.5

13 15

0.00241

Additionally, all the mixture parameters, UXY, WXYM, and Q1XYM, are calculated by eqs 19 and 27. The values obtained by Covington et al.4 together with our experimental results are then used to obtain UXY = −1.889, WXYM = 0.16 and Q1XYM = 0.2907 with σ = 0.0036. Table 7 represents the calculated mean PSC activity coefficients for KCl and KBr (γPSC ±KCl, γ±KBrl), osmotic PSC coefficients (φ ) and excess Gibbs free energies (Gex,PSC) based on the PSC approach. According to the standard deviation of the fit, the comparison of two models shows that the EP model presents a better fit of the experimental osmotic coefficients for the investigated system. A comparison of the two approaches which are depicted in Figure 5 confirms this assertion. Finally in this work the activity coefficients γ±, osmotic coefficient φ, and excess Gibbs free energy GEX of the ternary (KCl+KBr)(aq) as a function of ionic strengths in different ionic strength fractions, yB, by extended Pitzer method and Pitzer−Simonson−Clegg model are reported.

Table 5. Activity Coefficients γ±, Osmotic Coefficient φ, and Excess Gibbs Free Energy Gex of the Ternary (KCl + KBr)(aq) as a Function of Ionic Strengths I and Ionic Strength Fractions yB by Extended Pitzer (EP) Method Using Optimized Ternary Parameters (See Text) and Binary Parameters Taken from refs 8 and 14 at T = 298.15 K (yB = 0.25)

(yB = 0.51)

(yB = 0.75)

I/mol·kg−1

ϕEP

γEP ±KCl

γEP ±KBr

Gex

I/mol·kg−1

ϕEP

γEP ±KCl

γEP ±KBr

Gex

I/mol·kg−1

ϕEP

γEP ±KCl

γEP ±KBr

Gex

0.3204 0.4836 0.6064 0.8528 0.9560 1.0028 1.5016 1.9892 2.5404 3.0544 3.5444 4.0084 4.5220 4.8028 4.8900

0.9076 0.9038 0.9029 0.9038 0.9049 0.9054 0.9139 0.9249 0.9390 0.9532 0.9672 0.9807 0.9956 1.0037 1.0062

0.6817 0.6523 0.6369 0.6156 0.6091 0.6064 0.5876 0.5791 0.5760 0.5773 0.5809 0.5861 0.5934 0.5979 0.5994

0.6845 0.6562 0.6416 0.6217 0.6158 0.6134 0.5970 0.5905 0.5895 0.5924 0.5975 0.6039 0.6124 0.6175 0.6191

−457.0 −783.1 −1047.7 −1613.1 −1860.1 −1973.7 −3228.7 −4501.9 −5960.2 −7317.3 −8595.7 −9785.1 −11071 −11759 −11970

0.3938 0.5071 0.6241 0.7465 0.9367 1.1031 1.2524 1.5194 1.8378 2.4281 3.4087 4.0781 4.5983 4.9513

0.9042 0.9021 0.9012 0.9012 0.9024 0.9043 0.9064 0.9112 0.9178 0.9322 0.9593 0.9788 0.9942 1.0046

0.6662 0.6483 0.6343 0.6228 0.6092 0.6004 0.5941 0.5859 0.5796 0.5747 0.5781 0.5853 0.5928 0.5987

0.6692 0.6520 0.6386 0.6277 0.6152 0.6071 0.6015 0.5944 0.5894 0.5867 0.5931 0.6023 0.6113 0.6180

−601.8 −836.5 −1092.7 −1372.5 −1825.9 −2236.8 −2614.2 −3304.5 −4145.5 −5730.1 −8360.8 −10116 −11444 −12323

0.4137 0.4695 0.6329 0.8635 0.9729 1.1463 1.7696 2.1119 2.5552 3.1477 3.7004 4.0377 4.6580 4.8295

0.9028 0.9016 0.8998 0.9000 0.9008 0.9026 0.9134 0.9211 0.9320 0.9480 0.9636 0.9734 0.9916 0.9966 1.0008

0.6621 0.6531 0.6327 0.6131 0.6062 0.5974 0.5794 0.5750 0.5729 0.5743 0.5788 0.5826 0.5914 0.5941 0.5965

0.6649 0.6562 0.6365 0.6180 0.6116 0.6035 0.5879 0.5848 0.5841 0.5875 0.5937 0.5986 0.6093 0.6126 0.6155

−644.1 −759.7 −1117.4 −1657.8 −1925.3 −2359.9 −3996.7 −4923.2 −6134.6 −7753.7 −9246.9 −10145 −11762 −12200 −12565

F


ϕPSC

0.9085 0.9042 0.9031 0.9035 0.9044 0.9049 0.9129 0.9239 0.9383 0.9529 0.9676 0.9818 0.9979 1.0067 1.0094

I/mol·kg−1

0.3204 0.4836 0.6064 0.8528 0.9560 1.0028 1.5016 1.9892 2.5404 3.0544 3.5444 4.0084 4.5220 4.8028 4.8900

0.6844 0.6545 0.6387 0.6166 0.6098 0.6071 0.5871 0.5777 0.5740 0.5747 0.5781 0.5832 0.5904 0.5950 0.5965

γPSC ±KCl

(yB = 0.25) 0.6846 0.6551 0.6399 0.6188 0.6124 0.6099 0.5921 0.5848 0.5834 0.5862 0.5916 0.5984 0.6075 0.6131 0.6149

γPSC ±KBr −456.7 −784.9 −1052.2 −1624.9 −1875.7 −1991.2 −3269.3 −4569.3 −6060.5 −7448.8 −8756.5 −9972.1 −11284 −11986 −12201

Gex 0.3938 0.5071 0.6241 0.7465 0.9367 1.1031 1.2524 1.5194 1.8378 2.4281 3.4087 4.0781 4.5983 4.9513

I/mol·kg−1 0.9062 0.9038 0.9027 0.9024 0.9032 0.9048 0.9067 0.9110 0.9173 0.9312 0.9580 0.9777 0.9935 1.0043

ϕPSC 0.6714 0.6535 0.6394 0.6277 0.6139 0.6047 0.5983 0.5896 0.5829 0.5771 0.5793 0.5860 0.5933 0.5990

γPSC ±KCl

(yB = 0.51) 0.6665 0.6482 0.6337 0.6219 0.6080 0.5989 0.5926 0.5843 0.5781 0.5737 0.5786 0.5874 0.5962 0.6031

γPSC ±KBr −601.5 −837.7 −1096.2 −1379.1 −1838.5 −2255.7 −2639.6 −3343.0 −4201.7 −5823.7 −8524.9 −10331 −11698 −12604

Gex 0.4137 0.4695 0.6329 0.8635 0.9729 1.1463 1.7696 2.1119 2.5552 3.1477 3.7004 4.0377 4.6580 4.8295

I/mol·kg−1 0.9060 0.9048 0.9028 0.9027 0.9033 0.9048 0.9145 0.9216 0.9321 0.9475 0.9629 0.9726 0.9910 0.9961 1.0004

ϕPSC 0.6699 0.6611 0.6409 0.6214 0.6145 0.6057 0.5874 0.5828 0.5804 0.5817 0.5862 0.5901 0.5992 0.6021 0.6046

γPSC ±KCl

(yB = 0.75) 0.6599 0.6504 0.6287 0.6078 0.6003 0.5908 0.5710 0.5659 0.5632 0.5641 0.5683 0.5721 0.5810 0.5838 0.5863

γPSC ±KBr

−636.5 −751.1 −1106.1 −1643.6 −1910.1 −2343.6 −3980.3 −4909.2 −6125.8 −7754.6 −9259.0 −10165 −11796 −12238 −12606

Gex

Table 7. Activity Coefficients γ±, Osmotic Coefficient φ, and Excess Gibbs Free Energy Gex of the Ternary (KCl+KBr)(aq) as a Function of Ionic Strengths I and Ionic Strength Fractions yB by Pitzer−Simonson−Clegg (PSC) Method Using Optimized Ternary Parameters (See Text) and Binary Parameters Taken from refs 13 and 15 at T = 298.15 K

Journal of Chemical & Engineering Data Article

G



Article

(3) McCoy, W. H.; Wallace, W. E. Activity Coefficients in Concentrated Aqueous KCl-KBr Solutions at 25. J. Am. Chem. Soc. 1956, 78, 1830−1833. (4) Covington, A. K.; Lilley, T. H.; Robinson, R. A. Excess Free Energies of Aqueous Mixtures of Some Alkali Metal Halide Salt Pairs. J. Phys. Chem. 1968, 72, 2759−2763. (5) Rard, J. A, Platford, R. A Activity Coefficients in Aqueous Solutions, 2nd ed.;Pitzer, K. S, Ed.; CRC: Boca Raton, FL. 1991 Chapter 5. (6) Hefter, G.; May, P. M.; Marshall, S. L.; Cornish, J.; Kron, I. Improved apparatus and procedures for isopiestic studies at elevated temperatures. Rev. Sci. Instrum. 1997, 68, 2558−2567. (7) Archer, D. G. Thermodynamic Properties of the NaCl+H2O System. II. Thermodynamic Properties of NaCl(aq), NaCl·2H2O(cr), and Phase Equilibria. J. Phys. Chem. Ref. Data 1992, 21, 793−829. (8) Archer, D. G. Thermodynamic Properties of the KCl+H2O System. J. Phys. Chem. Ref. Data 1999, 28, 1−17. (9) Clegg, S. L.; Rard, J. A.; Pitzer, K. S. Thermodynamic properties of 0−6 mol kg mol−1 aqueous sulfuric acid from 273.15 to 328.15 K. J. Chem. Soc., Faraday Trans. 1994, 90, 1875−1894. (10) Pitzer, K. S.; Simonson, J. M. Thermodynamics of multicomponent, miscible, ionic systems: theory and equations. J. Phys. Chem. 1986, 90, 3005−3009. (11) Simonson, J. M.; Pitzer, K. S. Thermodynamics of multicomponent, miscible ionic systems: the system lithium nitratepotassium nitrate-water. J. Phys. Chem. 1986, 90, 3009−3013. (12) Clegg, S. L.; Pitzer, K. S. Thermodynamics of Multicomponent, Miscible, Ionic Solutions: Generalized Equations for Symmetrical Electrolytes. J. Phys. Chem. 1992, 96, 3513−3520. (13) Clegg, S. L.; Pitzer, K. S.; Brimblecombe, P. Thermodynamics of Multicomponent, Miscible, Ionic Solutions. 2. Mlxtures Includlng Unsymmetrical Electrolytes. J. Phys. Chem. 1992, 96, 9470−9479. (14) Popovic, D. Z.; Miladinovic, J.; Miladinovic, Z. P.; Grujic, S. R.; Todorovic, M. D.; Rard, J. Isopiestic measurements were made for {yKBr + (1 − y)K2HPO4}(aq) at T = 298.15 K. J. Chem. Thermodyn. 2013, 62, 52−161. (15) Hamer, W. J.; Wu, Y.-C. Osmotic Coefficients and Mean Activity Coefficients of Uni-univalent Electrolytes in Water at 25 °C. J. Phys. Chem. Ref. Data 1972, 1, 1047−1099. (16) Rard, J. A.; Miller, D. G. Isopiestic determination of the osmotic and activity coefficients of aqueous mixtures of NaCl and MgCl2 at 25 °C. J. Chem. Eng. Data 1987, 32, 85−92. (17) Robinson, R. A. Activity coefficients of sodium chloride and potassium chloride in mixed aqueous solutions at 25 °C. J. Phys. Chem. 1961, 65, 662−667.

Figure 5. Differences between the experimental osmotic coefficients Φexp and those calculated using (△) PSC and (∗) EP equations plotted against ionic strength.

5. CONCLUSION As a continuation of our previous study, the present investigation reports the isopiestic vapor pressures measuring electrolytic systems containing the ternary KBr + KCl + H2O system at 298.15 K with KCl(aq) serving as reference standard solution. Total molalities ranged from 0.3 up to 4.95 mol kg−1 in a series of mixed electrolyte systems characterized by their ionic strength fraction of KCl (yB = 0.25, 0.51, and 0.75). The modeling was conducted by the extended Pitzer and Pitzer− Simonson−Clegg equations. In these calculations all of the critically assessed parameters for the binary constituents are included, and hence the values of ΔΦ residuals are nearly random. In contrast to two previous KCl−KBr mixed salt isopiestic studies which were restricted to a small concentration range, the results of the present work are of a wider molalities range and of higher precisions as illustrated in Figure 3 and Figure 4. The previous results deviate from experimental data while the present results obtained with the use fitted parameters agrees consistently. Furthermore, on the basis of the standard deviation of our fits, the EP equation presents a better fit of the experimental osmotic coefficients for the investigated systems. Correspondingly activity coefficients γ±, osmotic coefficient φ, and excess Gibbs free energy GEX of the ternary (KCl +KBr)(aq) as a function of ionic strengths I and ionic strength fractions yB by extended Pitzer (EP) method and Pitzer− Simonson−Clegg were calculated.

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected]. ORCID

Zohreh Karimzadeh: 0000-0001-7140-8239 Funding

The authors would like to acknowledge the Azarbaijan Shahid Madani University. Notes

The authors declare no competing financial interest.

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REFERENCES

(1) Pitzer, K. S. Ed. Activity Coefficients in Electrolyte Solutions, 2nd ed.; CRC Press: Boca Raton, FL, 1991. (2) Salamat-Ahangari, R. Isopiestic investigation of the ternary system NaBr + NaCl + H2O at 298.15. J. Mol. Liq. 2016, 219, 1000− 1005. H


Determination of Activity Coefficients, Osmotic Coefficients, and

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