Thermodynamics of formation of double salts and mixed crystals from aqueous solutions

J. Chem. Thermodynamics 37 (2005) 1036–1060 www.elsevier.com/locate/jct

Thermodynamics of formation of double salts and mixed crystals from aqueous solutions Christomir Christov

*

Department of Chemistry and Biochemistry, University of California, San Diego, La Jolla, CA 92093-0340, USA Received 13 October 2004; received in revised form 7 January 2005; accepted 10 January 2005 Available online 3 March 2005

Abstract This paper describes a chemical model that calculates (solid + liquid) equilibria in the (m1CuCl2 + m2NiCl2)(aq), (m1MgCl2 + m2CrCl3)(aq), (m1(NH4)2SO4 + m2CuSO4)(aq), and (m1(NH4)2SO4 + m2MgSO4)(aq) systems, where m denotes molality at T = 298.15 K. The Pitzer ion-interaction model has been used for thermodynamic analysis of the experimental mixing solutions solubility data, presented in the literature. The thermodynamic functions needed have been calculated and the theoretical solubility isotherms have been plotted. The mixed solution model parameters {h(MN) and w(MNX)} have been chosen on the basis of the compositions of saturated ternary solutions and data on the pure water solubility of the (NH4)2SO4 Æ CuSO4 Æ 6H2O and (NH4)2SO4 Æ MgSO4 Æ 6H2O double salts. The results of Pitzer ion interaction model-based thermodynamic studies at T = 298.15 K on 62 binary, 82 ternary and eight multicomponent (water + salt) systems where solid phases with a constant stoichiometric composition (simple and double salts) crystallize have been summarized. The values of the pure electrolyte and mixing ion-interaction parameters, which give the best fit of the activity data in binary solutions and of the ternary solutions solubility data are tabulated. Important thermodynamic characteristics (thermodynamic solubility products K
sp ; standard molar Gibbs free energy of reaction Dr G
m of the synthesis of the double salts from the corresponding simple salts, components of the ternary solutions; and the standard molar Gibbs free energy of formation Df G
m ) of the solids crystallizing from the saturated binary and ternary solutions are given. The Df G
m values obtained are compared with those available in the literature. The results of thermodynamic simulation of binary and ternary systems at temperatures differing from the standard (273.15 K and 348.15 K) are presented. The model parameterization is tested by comparison of the predicted and the experimental composition of the invariant points in multicomponent solutions (not used in parameterization process). The component activities of the saturated (m1(NH4)2SO4 Æ CuSO4 + m2(NH4)2SO4 Æ MgSO4)(aq) and in the mixed crystalline phase were determined and the change of the molar Gibbs free energy of mixing Dmix G
m ðsÞ of crystals was determined as a function of the solid phase composition. It is established that at T = 298.15 K the mixed (NH4)2SO4 Æ (Cu,Mg)SO4 Æ 6H2O crystals show positive deviations from the ideal mixed crystals. Values for the integral Gibbs free energy of mixing Dmix G
m ðsÞ and excess Gibbs free energy of mixing DGEm ðsÞ in crystals calculated by different methods for the 23 ternary systems with co-crystallization of the salt components have been summarized. Ó 2005 Elsevier Ltd. All rights reserved. Keywords: Pitzer model; Ion interaction parameters; Solubility diagram; Simple salts; Double salts; Mixed crystals; Standard molar Gibbs free energy

1. Introduction Computer models that predict (solid + liquid + gas) equilibria close to experimental accuracy over broad ranges *

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0021-9614/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.jct.2005.01.008

of composition and temperature have wide applicability. They can simulate the complex changes that occur in nature and can replicate conditions that are difficult or expensive to duplicate in the laboratory. Such models can be powerful predictive and interpretive tools to study the geochemistry of natural waters and mineral deposits, solve environmental problems and optimize industrial processes.

C. Christov / J. Chem. Thermodynamics 37 (2005) 1036–1060

Pitzer created a method [1,2] that allows the calculation of the activity coefficients in unsaturated solutions of electrolytes with an accuracy of 2% to 6% [3]. Pitzer and co-authors [3–5] determined the ion interaction parameters for a large number of binary and ternary solutions. The specific interaction approach for describing aqueous solutions to high concentration, introduced by Pitzer, represents a significant advance in physical chemistry that has facilitated the construction of accurate thermodynamic models. The small number of parameters taking into account the ion interactions even in highly concentrated multicomponent solutions and the not complicated fundamental equations allowing relatively easy computerization, have contributed to the model becoming a real hit in the scientific literature. The Pitzer model is used successfully for the solution of various theoretical and practical tasks. Harvie and Weare [6] showed that the Pitzer specific interaction approach for describing aqueous solutions to high concentration could be expanded to calculate accurately mineral solubilities in complex brines. Weare and coworkers successfully applied their models at T = 298.15 K to the interpretation of natural and industrial processes (e.g., Harvie and Weare [6]; Harvie et al. [7]; Felmy and Weare [8]). Since 1980, using the same approach, many chemical models of (solid + liquid) equilibria in multicomponent brines have been developed that incorporate the excess Gibbs free energy equations of Pitzer. These models extend from subzero temperatures (Spencer et al. [9]; Marion [10]) to high temperatures below T = 623.15 K (e.g., Monnin and Schott [11]; Pabalan and Pitzer [12,13]; Moller [14]; Greenberg and Moller [15]; Millero and Pierrot [16]; Koenigsberger et al. [17]; Christov and Moller [18–20]; Weare et al. [21]). The applicability of Pitzer equations to the simulation of solutions from which solid solutions crystallize was also demonstrated [22]. In this article, a chemical equilibrium model is described that accurately calculates (liquid + solid) equilibria in the (m1CuCl2 + m2NiCl2)(aq), (m1MgCl2 + m2CrCl3)(aq), (m1(NH4)2SO4 + m2CuSO4)(aq), and (m1(NH4)2SO4 + m2MgSO4)(aq) systems, at T = 298.15 K. The mixing solution model for sulfate systems has been used for thermodynamic simulation of (m1 (NH4)2SO4 Æ CuSO4 + m2(NH4)2SO4 Æ MgSO4)(aq) system and determination of the molar Gibbs free energy of mixing Dmix G
m ðsÞ of (NH4)2SO4 Æ (Cu,Mg)SO4 Æ 6H2O crystals as a function of the solid phase composition. The main purpose of the present paper is to summarize and specify the published results of the author [23–65], obtained using Pitzer ion-interaction model in thermodynamic studies on different type of binary, ternary, and multicomponent (water + salt) systems. Important thermodynamic functions (pure electrolyte and mixing ion-interaction parameters and thermodynamic solubility product of solid phases) needed for constructing the

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solubility model of ternary and multicomponent systems and for determining the thermodynamic characteristics of the formation process of mixed crystals have been tabulated. The values of the standard molar Gibbs free energy of formation ðDf G
m Þ for 66 simple and 62 double salts crystallizing from saturated binary and ternary solutions studied by the author, have been calculated. The results from thermodynamic studies of 23 ternary systems with isomorphic or isodimorphic co-crystallization of the salt components have been summarized. The equilibrium models summarized here have important theoretical and practical applications. The parameterization presented can play significant roles in the behaviour of many natural and industrial fluids. Thus, the model of iron (II) and iron (III), Al(III), Cr(III) equilibria in aqueous chloride and sulfate solutions forms a useful base for developing comprehensive models that calculate speciation and solubilities as a function of temperature and composition over wide pH ranges. The study of the quarternary {NaCl + Na2SO4 + Na2Cr2O7 + H2O}, {KCl + K2SO4 + K2Cr2O7 + H2O}, and (m1M2SO4 + m2M 0 SO4) (aq) (M = Na, K, NH4; M 0 = Al, Cr) is of great practical importance above all in association with the choice of optimum preparation conditions of sodium-, and potassium-bichromate, alums, and chromium alums, respectively. The investigation of ternary and multicomponent solutions with participation of chloro-carnallites and bromo-carnallites [MX Æ MgX2 Æ 6H2O(s); (M = Li, K, NH4, Rb, Cs; X = Cl, Br)] is of a practical importance with a view to explaining the distribution of alkali metal-, chloride-, and bromide-ions in natural evaporate deposits, during the crystallization of salts as a result of seawater evaporation, and during the treatment of natural carnallite deposits. 2. Theoretical background and modeling approach 2.1. Pitzer equations Harvie and Weare [6] showed that the Pitzer approach could be expanded to calculate accurately solubilities in complex brines. In this study, we summarize the solubility models that incorporate the ionic strength-dependent specific interaction equations for estimating the osmotic coefficient and activity coefficients of ions in electrolyte solutions as described in references [6,7]. These equations have been discussed in detail in many other publications [8,16]. Here, only the expression for the activity coefficient of the hydrogen ion (H) ðcðHþ Þ Þ is given X cðHþ Þ ¼ Z 2H F þ ma ð2BHa þ ZC Ha Þþ a
 X X mc 2UHc þ ma wHca þ c X X XX ma ma0 waa0 H þ jzH j mc ma C ca : ð1Þ a
c

a

1038

C. Christov / J. Chem. Thermodynamics 37 (2005) 1036–1060

Equation (1) is symmetric for cations and anions. The subscripts c and a refer to cations and anions, and m is their molality. The double summation index, a < a 0 , denotes the sum over all distinguishable pairs of dissimilar anions. B and U represent measurable combinations of the second virial coefficients. C and w represent measurable combinations of third virial coefficients. B and C are parameterized from single electrolyte data, and U and w are parameterized from mixed solution data. The function F is the sum of the Debye–Hu¨ckel term      Au I 1=2 1 þ bI 1=2 þ ð2=bÞ ln 1 þ bI 1=2 ; ð2Þ and terms with the derivatives of the second virial coefficients with respect to ionic strength [7]. In equation (2), b is a universal empirical constant assigned to be equal to 1.2. Au (Debye–Hu¨ckel limiting law slope for osmotic coefficients) is a function of temperature, density and dielectric constant of water [14,18]. For the interaction of any cation M and any anion X, Pitzer assumes that in equation (1) B has the ionic strength-dependent form   ð0Þ ð1Þ  ð2Þ  ð3Þ BMX ¼ bMX þ bMX g a1 I 1=2 þ bMX g a2 I 1=2 ; where gðxÞ ¼ 2½1  ð1 þ xÞ expðxÞ=x2 with x = a1I1/2 or a2I1/2. a is a function of electrolyte type and does not vary with concentration and temperature. In the Harvie and Weare [6] approach, when either cation or anion for the electrolyte considered is univalent a1 is equal to 2.0 and the b(2) term is omitted. For 2-2 or higher valence pairs, a1 equals 1.4 and a2 equals 12 [5,6]. The modified form, assigned by Pitzer, for these higher valence species accounts for their increased tendency to associate in solution. In order to enlarge the molality range of application of single electrolyte parameters for MCl2(aq) (M = Ni, Cu, Co, Mn) (electrolytes of type 2-1) and HCl(aq) (1-1 electrolyte) Filippov et al. [68,69] used b(2) term. The standard ion interaction model does not use b(2) term for 2-1 and 1-1 electrolytes, although its inclusion makes it possible to fit aqueous solution properties to very high concentrations {see Pabalan and Pitzer [70]}. To fit the activity data in unsaturated Al2(SO4)3(aq) (up to 1.00 mol Æ kg1) Reardon set: b(2) =  500, a1 = 2.0, and a2 = 50 [71]. In our modeling approach, in order to (i) enlarge the molality range of application of single electrolyte parameters and to (ii) reduce the standard deviation of fit (r) (for definition of sigma, see Christov and Moller [19]) of activity data used in parameterization, we have varied a1 and a2 values (see the text in next section and table 1). The CMX third virial coefficients (see equation (1)) are assumed to be ionic strength independent. However, some terms containing CMX parameters have a concentration dependence given by Z

Z¼

X

mi jzi j;

ð4Þ

where m is the molality of species i and z is its charge. The U terms account for interactions between two ions i and j of like charges. In the activity coefficient expression (equation (1)) U ¼ hij þ E hij ðIÞ:

ð5Þ

In equation (5), hij is the only adjustable parameter. The term Ehij(I) accounts for electrostatic asymmetric mixing effects that depend only on the charges of ions i and j and the total ionic strength. Note that in constructing the mixing solution models, summarized in this study, we used unsymmetric mixing term Ehij. The wijk parameters, which are used for each triple ion interaction where the ions are all not of the same sign, are assumed to be independent of ionic strength. Their inclusion is important for describing solubilities in concentrated multicomponent systems. Therefore, at constant temperature and pressure, the solution model parameters to be evaluated are b(0), b(1), b(2), and Cu for each cation–anion pair; h for each unlike cation–cation or anion–anion pair; w for each triple ion interaction where the ions are all not of the same sign. The systems presented here do not include neutral species, therefore neutral species – ion interaction parameters (k and n 0 : see Felmy and Weare [8]) are not introduced in our models. To extend the application of the model for predictions of solution properties at very high concentrations Pitzer and Simonson [72] introduced in their approach mole fraction concentrations. Their approach has been used successfully for modeling CaCl2(aq) {Rard and Clegg [73]} and aerosol mixtures {Clegg et al. [74]}. In the parameterization presented here, we used a standard molality-based ion interaction model. To fit the binary solution properties with lower r value, some authors used in their single electrolyte parameterization an extended version of Pitzer model, introducing two Cu parameter values {see Palmer et al. [75]}. In our single electrolyte parameterization, we used standard Pitzer approach with one Cu parameter. Note that in the model presented, we accepted that the electrolytes are completely dissociated and there are only independent ions in the solution. The equilibrium constant of complexes (such as HSO 4; þ AlSOþ ; FeSO ) are not included in the model. 4 4 3. Model parameterization In the model presented, we used the third virial coefficient structure of the Pitzer excess Gibbs free energy expansion. Therefore, only binary and ternary data are required to parameterize fully a model of higher order systems [7,18]. The complex systems investigated by us can, on the whole, be divided into two main groups:

C. Christov / J. Chem. Thermodynamics 37 (2005) 1036–1060

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TABLE 1 Pitzer single electrolyte parameters at T = 298.15 K where r is the standard deviation of the osmotic coefficientsa No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62

System b

HNO3 + H2O LiCl + H2O LiBr + H2O Li2SeO4 + H2Oc NaOH + H2O NaCl + H2O NaCl + H2O NaBr + H2O Na2CO3 + H2O Na2SO4 + H2O Na2SO4 + H2O Na2SeO4 + H2O Na2CrO4 + H2O Na2Cr2O7 + H2Ob NH4Cl + H2O NH4Br + H2O (NH4)2SO4 + H2O (NH4)2SeO4 + H2O KCl + H2O KBr + H2O KI + H2O K2CO3 + H2O K2SO4 + H2O K2SeO4 + H2O K2CrO4 + H2O K2Cr2O7 + H2Oe RbCl + H2O RbBr + H2O RbI + H2O Rb2SO4 + H2O Rb2SeO4 + H2O CsCl + H2O CsBr + H2O Cs2SO4 + H2O Cs2SeO4 + H2O MgCl2 + H2O MgBr2 + H2O MgSO4 + H2O MgSO4 + H2O MgSeO4 + H2Od CaCl2 + H2O CaBr2 + H2O CaSO4 + H2O MnCl2 + H2Oc MnBr2 + H2O CoCl2 + H2Oc CoSO4 + H2O CoSeO4 + H2Od NiCl2 + H2Oc NiCl2 + H2O NiBr2 + H2O NiSO4 + H2O NiSeO4 + H2Od CuCl2 + H2Ob CuSO4 + H2O ZnSO4 + H2O ZnSeO4 + H2Od AlCl3 + H2O Al2(SO4)3 + H2Ob FeCl2 + H2Ob FeCl3 + H2Ob FeSO4 + H2O

b(0)

b(1)

0.0872 0.20972 0.24554 0.24682 0.17067 0.0765 0.0765 0.11077 0.05306 0.04604I 0.01958 0.12380 0.0645 0.13513 0.0521 0.0618 0.04841 0.04155 0.04835 0.0559 0.0658 0.20366 0.04995 0.10911 0.0791 2.1239 0.0409 0.0370 0.0427 0.06518 0.12672 0.0390 0.0301 0.09484 0.18234 0.3511 0.4328 0.2210 0.215 0.3010 0.31590 0.33899 0.2000 0.33721 0.44655 0.34723 0.1631 0.4409 0.34689 0.39304 0.44305 0.1702 0.58712 0.17661 0.234 0.1949 0.2324 0.71532 0.56622 0.40942 0.34082 0.2568

0.1186 0.34380 0.44244 0.53473 0.08411 0.2665 0.2664 0.13760 1.29262 0.93350 1.113 0.30925 1.5974 8.94435 0.1916 0.1635 1.13240 0.77095 0.21220 0.2296 0.3064 1.39843 0.77930 0.71354 1.1130 6.3443 0.1919 0.1520 0.1107 0.7407 0.25052 0.0374 0.0029 0.6026 1.64961 1.6512 1.7457 3.343 3.365 4.2720 1.614 2.04551 3.7762 1.50854 1.34477 1.66470 3.346 1.4440 1.59327 0.99773 1.48323 2.9070 2.98530 0.57402 2.527 2.883 3.8156 5.65087, 12.16131 1.99612 1.6285 3.063

b(2) 0.1555

0.00621

2.03150



25.9183

37.23 32.74

58.388 0.00078 0.00350 30.70 0.00275

40.06 0.63405 48.33 32.81

3.07519 0.34439 1.7199 42.0

Cu

mmax/(mol Æ kg1)

r

Reference

Activity data source

0.0023 0.00433 0.00293 0.04756 0.00342 0.00062 0.00127 0.00153 0.00094 0.00483 0.00497 0.00729 0.0094 0.00495 0.0030 0.0042 0.00155 0.00312 0.00084 0.0017 0.0022 0.00649

29.3 19.219 20.000 3.708 29.00

0.0043 0.05339 0.09391 0.00038 0.08591 0.002 0.001 0.00448 0.00257 0.00112 0.003 0.00595 0.00348 0.00268

[23] [80] [80] [36] [80] [69] [4,6,7] [80] [80] [80] [6,7] [37] [58] [24] [27] [27] [80] [26] [6,7] [27] [45] [52] [7] [26] [58] [40] [27] [27] [45] [75] [61] [27] [27] [75] [56] [27] [27] [5,7] [82] [51] [6] [80] [80] [29] [80] [29] [84,85] [51] [29] [80] [80] [5] [51] [29] [81] [5] [51] [38] [38] [65] [65] [83]

[86]

0.00325 0.0012 1.6682 0.0007 0.0007 0.0016 0.0006339 0.00860 0.0012 0.0005 0.0002549 0.01407 0.0065 0.0029 0.025 0.028 0.0163 0.00034 0.01067 0.02724 0.02269 0.00559 0.037 0.0303 0.00801 0.01658 0.00590 0.0366 0.04540 0.01089 0.0044 0.0299 0.0224 0.00418 0.000524 0.02643 0.0140 0.0209

6.00 9.00 2.750 1.750 4.00 2.811 5.07 3.879 7.06 7.63 5.500 5.502 4.800 5.67 8.56 7.97 0.700 4.742 3.37 0.507 7.51 6.40 7.72 2.048 5.20 11.30 5.67 3.1082 4.591 5.76 5.37 3.07 3.28 2.5 6.0 0.02 8.13 5.64 4.26 2.39 2.82 4.98 5.50 4.50 2.50 1.73 5.73 1.4 3.50 3.05 3.44 1.118 4.90 6.07

0.0028 0.00185 0.00351 0.0005 0.0010 0.00290 0.002 0.00698 0.00063 0.00188 0.0010 0.0020 0.00336 0.0030 0.0010 0.00105 0.0041 0.0055

0.005 0.003 0.00715 0.00460 0.00093 0.00546 0.00079 0.0045 0.00095 0.0139 0.00866 0.005 0.00243 0.0021310 0.003 0.0056 0.00336 0.00125 0.00159 0.00218

[36]

[37] [86] [24] [86] [86] [26] [86] [86] [86] [26] [87] [87] [86] [86] [86] [61] [86] [86] [56] [86] [86]

[88]

[86] [86] [88] [86]

[88] [86]

[88] [86] [86] [86] [86] (continued on next page)

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C. Christov / J. Chem. Thermodynamics 37 (2005) 1036–1060

TABLE 1 (continued) No.

System

b(0)

b(1)

b(2)

Cu

mmax/ (mol Æ kg1)

r

Reference

Activity data source

63 64 65 66

Fe2(SO4)3 + H2Of CrCl3 + H2O Cr2(SO4)3 + H2Ob KCr(SO4)2 + H2Ob

0.56622 0.72234 0.44794 1.3750

12.16131 5.59892 6.19731 27.8819

3.07519

0.000524 0.04141 0.02024 0.36949

1.118 1.4 1.4 0.8

0.00125 0.00388 0.00065 0.00110

[65] [62] [59] [59]

[86] [90] [89,90] [90]

4.86956 9.31138

a Parameters No. 6 have been used for simulation of (m1NaCl + m2CuCl2)(aq) and {NaCl + CuCl2 + Na2SO4 + CuSO4 + H2O} (variant I) systems. For all other ternary and quaternary systems with participation of NaCl the parameters No. 7 have been used. Parameters No. 10 have been used for simulation of (m1Na2SO4 + m2CuSO4)(aq) and {NaCl + CuCl2 + Na2SO4 + CuSO4 + H2O} (variant I) systems. For all other ternary and quaternary systems with participation of Na2SO4 we have used parameters No. 11. Parameters No. 39 were used in constructing of the (m1MgSO4 + m2Al2(SO4)3)(aq) mixing solution model. For all other ternary systems with participation of MgSO4 we used parameters No. 38. Parameters No. 50 have been used for simulation of (m1CsCl + m2NiCl2)(aq) system. For all other ternary systems with participation of NiCl2 we have used parametrs No. 49. b Calculated using a1 = 2 and a2 = 1. c Calculated using a1 = 2 and a2 = 1. d Calculated using a = 1.4. e Calculated using a1 = 1.4 and a2 = 12. f Pure electrolyte Al2(SO4)3(aq) parameters from Christov [57,59] have been used in constructing the mixed solutions {m1FeSO4 + m2Fe2(SO4)3}(aq) and {m1K2SO4 + m2Fe2(SO4)3}(aq) model {valid up to 1 mol kg1 of Fe2(SO4)3} [65].

(I) ternary and multicomponent systems where phases with a constant stoichiometric composition (simple and double salts) crystallize and (II) ternary systems with isomorphic and isodimorphic co-crystallization of components, i.e., systems in which continuous or discontinuous series of mixed crystals are formed. The main purposes of the simulation of the first group of systems are: (a) constructing (solid + liquid) equilibria models that give the best fit of the experimental solubility data and (b) determining the thermodynamic characteristics of the stoichiometric solid phases precipitating from the saturated ternary and quaternary solutions. The simulation of the second group of systems was aimed at determining the standard molar Gibbs free energy of mixing, Dmix G
m ðsÞ, and the excess Gibbs free energy of mixing, DGEm ðsÞ, in crystals and establishing their dependence on the composition of the mixed crystalline phase. Here, a new mixing solution model for both groups of systems are described. According to the experimental solubility data, the mixed (m1CuCl2 + m2NiCl2)(aq) [76] and (m1MgCl2 + m2CrCl3)(aq) [66] systems are of simple eutonic type at T = 298.15 K (i.e., only simple salts, components of the system, crystallize from saturated ternary solutions) (group (I)). From saturated ternary (m1(NH4)2SO4 + m2CuSO4)(aq) [77], and (m1(NH4)2SO4 + m2MgSO4)(aq) [78] solutions crystallize double salts with 1-1-6 stoichiometric composition (group (I)). According to the data of Hill et al. [79] solid solutions between (NH4)2SO4 Æ CuSO4 Æ 6H2O(s) and (NH4)2SO4MgSO4 Æ 6H2O are in equilibrium with saturated (m1(NH4)2SO4 Æ CuSO4 + m2(NH4)2SO4 Æ Mg(SO4)3)(aq) solutions at T = 298.15 K (group (II). 3.1. Solubility calculations The systems with crystallization of stoichiometric salts were investigated according to the following

scheme: the determination of Pitzer single electrolyte parameters (b(0), b(1), b(2), C/) for binary solutions; the evaluation of Pitzer mixing parameters (h and w) characterizing the interaction between two different ions of the same sign and the interaction between three ions, respectively; the calculation of the solubility isotherm of the ternary (m1CuCl2 + m2NiCl2)(aq), (m1MgCl2 + m2CrCl3)(aq), (m1(NH4)2SO4 + m2CuSO4)(aq), and (m1(NH4)2SO4 + m2MgSO4)(aq) systems; and, the calculation of the thermodynamic characteristics of the (NH4)2SO4 Æ CuSO4 Æ 6H2O(s) and (NH4)2SO4MgSO4 Æ 6H2O double salts, precipitating from saturated ternary sulfate solutions. 3.1.1. Binary systems The all binary parameters needed for construction of the models of mixing solutions noted above are taken from the literature (table 1). The pure electrolyte parameters for CuCl2(aq), NiCl2(aq), MgCl2(aq), CrCl3(aq), (NH4)2SO4(aq), CuSO4(aq), and MgSO4(aq) solutions at T = 298.15 K have been determined by many authors [5–7,27,29,62,80,81]. Since the calculation of the compositions of saturated ternary solutions is one of the main purposes of the simulation, the applicability of the binary parameters to high molality binary solutions up to saturation at the lowest value of the standard deviation (r) is a very important criterion for the choice of the binary parameters. For all chloride binary systems, the parameters evaluated in previous studies of the author are used [27,29,62] (table 1). The applicability of these parameters for describing the solution properties to high concentrations has been proved by constructing the solubility calculation models for different ternary and multicomponent systems, for which the binary CuCl2(aq), NiCl2(aq), MgCl2(aq), and CrCl3(aq) are sub-systems [27–29,34,43,44,57]. Note that the parameters of MgCl interactions (taken from reference [27]) are valid up

C. Christov / J. Chem. Thermodynamics 37 (2005) 1036–1060

to higher molality (almost up to saturation of MgCl2)(aq) than those recommended by Pitzer and Mayorga [5] and used in the T = 298.15 K model of Harvie et al. [7]. The pure electrolyte Mg-SO4 interaction parameters are taken from reference [5]. These parameters have been validated by (solid + liquid) equilibria calculations in comprehensive mineral solubility models at T = 298.15 K [6,7]. The (NH4)2SO4(aq) pure electrolyte parameters of Kim and Frederick [80] (mmax = 5.5 mol Æ kg1) are used in this study in constructing mixing ammonium sulfate solution models. Their applicability for solubility calculations has been demonstrated by the author of the present study in thermodynamic models for many ternary [59,60] and quaternary [60,63] systems. The CuSO4(aq) parameters are taken from Downes and Pitzer [81]. These parameters were utilized for (solid + liquid) equilibria calculations in the quaternary reciprocal Na-Cu-Cl-SO4-H2O [41], as well as in (m1K2SO4 + m2CuSO4)(aq) [44], and (m1CuSO4 + m2Al2SO4)3)(aq) [60] ternary systems. Table 1 summarizes the values of Pitzer single electrolyte parameters both: (i) evaluated by us and (ii) taken from the literature [5–7,69,75,80–85], which have been used in construction of (solid + liquid) equilibria model for complex solutions. In our binary solution parameterizations we used the data on the dependence of the activity of water aw, and/or of the osmotic coefficient /, and/ or mean activity coefficient c, on the molality m of the solutions. In the evaluation of pure electrolyte parameters, we used initial activity data for binary solutions only, i.e., without any complex systems solubility or activity data adjustments. This approach was accepted, because binary solution parameters are intimately interlinked with mixing model parameters. To solve the coupling of binary and mixing parameters, it is necessary to use in parameterization initial experimental data, which are related directly to single electrolyte parameters (more discussion on this problem is presented in section 3.1.3 and in reference [20]). Note also that the pure electrolyte parameters, evaluated by us are valid to the highest molality of the binary solutions and/or have a sigma (r) value lower than those of other sets of parameters presented in the literature (mmax in table 1 give a highest molality of initial experimental activity data, used in parameter evaluation). Therefore, the binary ion interaction parameters, presented in table 1, described the binary solutions activity data with a very good accuracy, and can be used for construction of mixing solution (solid + liquid) equilibria models. With the exceptions of the HNO3(aq), NaOH(aq), Na2CO3(aq), K2CO3(aq), and CaSO4(aq) parameters (see the text below for their application), the concentration extrapolation (up to saturation) of the all parameters presented in table 1 have been validated by constructing the mixing models, which accurately calculate the solubility, and/or component activities in saturated ternary and multicomponent solu-

1041

tions (see sections 3.1.2 and 3.1.3). The last column of table 1 gives the source of the experimental activity data used in our pure electrolyte parameter evaluations [86– 91]. There are no activity data for unsaturated Fe2 (SO4)3(aq) solutions. Note that in constructing the mixed solutions {m1FeSO4 + m2Fe2(SO4)3}(aq) and {m1K2SO4 + m2Fe2(SO4)3}(aq) model [65] the pure electrolyte Al2(SO4)3(aq) parameters from references [57,59] are accepted as an approximation for ferric sulfate binary parameters {valid up to 1 mol kg1 of Fe2(SO4)3}. Simulation of the binary solutions of alkali selenates and sodium bichromate was performed on the basis of experimental osmotic coefficients data obtained by us using the isopiestic method [24,26,36,37,56,61]. The comprehensive description of the isopiestic method for determination of aw, and/or / is given in references cited above. The new osmotic coefficient data for binary alkali bromide solutions, MgBr2(aq), CaBr2(aq), and FeSeO4(aq) systems over the temperature range (298.15 to 323.15) K, determined on the basis of the isopiestic measurements will be presented in future studies [66]. In order to reduce the r values and extend the concentration extrapolation of the ion interaction parameters up to saturation molality (ms), we used different approaches in our parameterization, varying the a1 and a2 values and introducing, or not, the b(2) parameter. The following schemes were used: (i) the approach with three parameters (b(0), b(1), and Cu), using a = 2 [1]; (ii) the standard approach for simulation of 2-2 electrolytes including b(2) parameter, with a1 = 1.4 and a2 = 12 [5]; (iii) the approach of Reardon [71] for simulation of 3-2 electrolytes Al2(SO4)3 including four parameters, based on the values a1 = 2 and a2 = 50 [71,70]; (iv) the approach of Filippov et al. [68,69] with four parameters (involving also b(2)) using a1 = 2 and a2 = 1; (v) the approach with four patameters, with a1 = 2 and a2 = 1 [68,29,32,36,44]; (vi) an approach with three parameterss (b(0), b(1), and Cu), using a = 1.4 [36,37,47,51]. The approach which give the best fit of experimental binary solution activity data is accepted (see table 1). Utilizing the system (HNO3 + H2O) by way of example, we proposed [23] an approach of application of the fundamental Pitzer equations to the determination of the thermodynamic characteristics of electrolytes with incomplete dissociation. The binary parameters presented in table 1 are used to determine the stoichiometric activity coefficient (cs) of HNO3 up to 29.3 mol Æ kg1, assuming complete dissociation of the acid. On the basis of Raman spectroscopy studies, the dependence of the degree of dissociation of nitric acid solutions on the molality (m) has been determinated. The data obtained and the thermodynamic approach of McKay [92] enabled us to calculate the activity coefficient of the ionized part (ci), the real activity coefficient (ch) and the thermodynamic dissociation constant K of nitric acid (K = 19.05) [23].

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C. Christov / J. Chem. Thermodynamics 37 (2005) 1036–1060

TABLE 2 Calculated values of the logarithm of the thermodynamic solubility product, K
sp , and of the standard molar Gibbs free energy of formation, Df G
m , of solid phases crystallizing from saturated binary solutions at T = 298.15 K Salt composition

ms/(mol Æ kg1)

ln K
sp

Df G
m =ðkJ  mol1 Þ Calculateda

Reference data From Wagman et al. [93]

19.81 17.21 4.275 28.53 6.16 9.19 2.77 1.97 3.03 5.30 7.106 7.45 8.02 5.806 6.715 4.762 5.75 8.90 7.97

11.91 11.60 4.058 11.679 3.63 4.661 1.889 2.834 1.59 2.32 2.42 2.86 3.066 0.57 2.118 2.058 2.60 4.00 13.614

632.14 842.77 1254.99 627.33 384.13 828.57 3427.59 3646.65 3340.34 2194.33 2293.17 203.45 175.67 901.74 594.7 409.39 380.78 324.92 1416.30

0.695 5.13 3.36 0.507 7.78 6.93 7.63 1.928

4.070 2.077 0.094 3.140 3.023 2.60 2.78 1.92

1321.16 1002.69 1294.06 1875.42 407.71 381.50 328.66 1317.25

5.20 11.37 5.79 5.00 6.34 5.86 5.56 3.026 3.28 7.32

0.99 3.49 1.905 1.958 1.45 10.60 12.21 4.29 2.608 9.46

1006.81 414.59 391.26 1323.72 1021.74 2113.75 2055.23 2869.87 2325.34 2215.36

CaBr2 Æ 6H2O CaSO4 Æ 2H2O MnCl2 Æ 4H2O MnBr2 Æ 4H2O CoCl2 Æ 6H2O Co(NO3)2 Æ 6H2O CoSO4 Æ 7H2O CoSeO4 Æ 6H2O NiCl2 Æ 6H2O NiBr2 Æ 6H2O Ni(NO3)2 Æ 6H2O NiSO4 Æ 7H2O

7.3 0.02 6.10 7.04 4.30 5.60 2.42 2.82 4.91 6.53 4.50 2.66

13.05 10.205 6.40 10.9 5.83 6.873 5.03 4.05 7.05 12.02 6.391 5.08

2151.93 1797.66 1423.21 1357.52 1725.18 1677.62 2471.30 1928.51 1713.35 1646.50 1670.01 2462.63

NiSeO4 Æ 6H2O CuCl2 Æ 2H2O Cu(NO3)2 Æ 6H2O CuSO4 Æ 5H2O Zn(NO3)2 Æ 6H2O ZnSO4 Æ 7H2O

1.93 5.68 8.04 1.46 6.72 3.59

3.18 4.47 6.682 6.01 7.963 4.277

1917.55 660.14 1558.20 1879.58 1767.58 2562.10

LiCl Æ H2O LiBr Æ 2H2O Li2SeO4 Æ H2O NaOH Æ H2O NaCl NaBr Æ 2H2O Na2CO3 Æ 10H2O Na2SO4 Æ 10H2O Na2SeO4 Æ 10H2O Na2CrO4 Æ 4H2O Na2Cr2O7 Æ 2H2O NH4Cl NH4Br (NH4)2SO4 (NH4)2SeO4 KCl KBr KI K2CO3 Æ 1.5H2O K2SO4 K2SeO4 K2CrO4 K2Cr2O7 RbCl RbBr RbI Rb2SO4 Rb2SeO4 CsCl CsBr Cs2SO4 Cs2SeO4 MgCl2 Æ 6H2O MgBr2 Æ 6H2O MgSO4 Æ 7H2O MgSeO4 Æ 6H2O CaCl2 Æ 6H2O

Other data

631.80 840.5 629.338 384.138 828.29 3427.66 3645.85

3428.428 [20]

2194.92 [95] 202.87 175.2 901.67 409.14 380.66 324.892 1432.5 1321.37 1002.8 1295.7 1881.8 407.80 381.79 328.86 1316.89

414.53 391.41 1323.58

1431.19 [7] 1434.49 [20]

1289.76 [97]

1317.165 [97] 1316.91 [75]

1323.82 [97]

2114.64 2055.7 2871.5 2215.2

2215.19 [7] 2214.64 [19]

2152.8 1796.28 1423.6 1725.2 2473.42

1677.43 [86] 2471.43 [70,84]

1713.19

2461.83

655.9 1879.745 1772.71 2562.67

2461.99 [97] 2462.73 [70,84] 660.44 [86] 1561.61 [86]

2562.36 [97]

C. Christov / J. Chem. Thermodynamics 37 (2005) 1036–1060

1043

TABLE 2 (continued) Salt composition

ms/(mol Æ kg1)

ln K
sp

Df G
m =ðkJ  mol1 Þ Calculateda

Reference data From Wagman et al. [93]

ZnSeO4 Æ 6H2O CrCl3 Æ 6H2O AlCl3 Æ 6H2O FeCl2 Æ 4H2O FeCl3 Æ 6H2O FeSO4 Æ 7H2O Al2(SO4)3 Æ 17H2O

3.05 4.50 3.38 5.09 6.22 1.92 1.125

3.62 13.60 16.05 6.97 6.64 5.13 14.11

Cr2(SO4)3 Æ 16H2O

1.634

12.21

a b

2020.11 1998.23 2261.67 1272.59 1804.70 2496.05 7269.76 7278.54b 6488.88

Other data

2261.1 1277.53 [86] 2496.00

2496.00 [70] 7279.22 [71]

Calculated using standard chemical potentials for ions in solutions from Wagman et al. [93]. Calculated using standard chemical potentials for Al3+(aq) from Robie et al. [98].

Using the data for b(0), b(1), b(2), and Cu presented in table 1, and the molality (ms) of the saturated binary solutions, we calculated the logarithm of the thermodynamic solubility product ln K
sp for the crystalline solid phases crystallizing from saturated binary solutions (table 2). The small differences between the ln K
sp values for some solid phases (e.g., KCl, RbCl, MgCl2 Æ 6H2O, CoCl2 Æ 6H2O, CuCl2 Æ 2H2O, NiSeO4 Æ 6H2O) presented in table 2 and those presented in references [6,29,34,47] are mainly due to the different ms values used for the calculation. On the basis of ln K
sp values and using standard chemical potentials for aqueous species from Wagman et al. [93], the standard molar Gibbs free energy of formation, Df G
m , of 63 simple salts, precipitating from saturated binary solutions have been determined. In table 2, there are compared the Df G
m values obtained in this study with those recommended by Wagman et al. [93] and presented in other references [71,83,86,94–97]. For most of the salts, the agreement obtained is excellent. With one exception (for K2CO3 Æ 1.5H2O(s)), the difference is less than 5 kJ Æ mol1. The new improved parameterization of K2CO3(aq), which give a very good agreement with Df G
m ðK2 CO  1:5H2 OðsÞÞ values presented in the literature is described in our carbonate (solid + liquid) equilibria model [20] (see table 2). The Df G
m ðAl2 ðSO4 Þ3  17H2 OðsÞÞ, calculated using Df G
m ðAl3þ ðaqÞÞ from Robie et al. [98] is in very good agreement with the value proposed by Reardon [71] (table 2). In reference [25], the ion interaction model has been used for the solution of an applied task: determination of the optimum (from a thermodynamic viewpoint) concentrations of binary and ternary nitrate solutions from which a Co, Ni, Cu or Zn active phase is deposited during catalyst preparation. The binary parameters are taken from Kim and Frederick [80]. The calculated ln K
sp and Df G
m values of M(NO3)2 Æ 6H2O(s) (M = Co, Ni, Cu or Zn) are given in table 2. The ternary nitrate systems are simulated, neglecting the ternary interactions in the solutions (i.e., accepting h = w = 0). The simulation has allowed prediction of the concentration intervals in which the activity coefficients of the

electrolytes have their lowest values and which is thermodynamically most favourable for adsorption from solutions. The predicted optimum concentration intervals have been confirmed experimentally [99]. In previous studies [52,53], the ion-interaction model has been used to predict the conversion of BaSO4 and CaSO4 to BaCO3 and CaCO3, respectively, in the quaternary (water + salt) systems BaSO4 þ Na2 CO3 ¼ BaCO3 þ Na2 SO4

ð6Þ

and CaSO4 þ Na2 CO3 ¼ CaCO3 þ Na2 SO4

ð7Þ

ln K
sp

values of the crystalline hydrates Using the formed in the saturated binary solutions (table 2), the following predictions are made: 1. In the presence of a ‘‘neutral’’ component (NaOH), system (6) would have a behaviour close to that of system BaSO4 + K2CO3 = BaCO3 + K2SO4, for which a BaSO4 to BaCO3 conversion has been established experimentally [52]; 2. In system (7), CaSO4 should dissolve, and CaCO3 precipitate, i.e., conversion to CaCO3 is to be expected [53]. These predictions have been confirmed experimentally. A conversion of 88% has been achieved of BaSO4 [52], and 98% of CaSO4 [53].

3.1.2. Ternary systems The Pitzer mixing ion interaction parameters (h and w) for the ternary systems under study have been evaluated on the basis of the experimental data on the compositions of the saturated ternary solutions. The mixed solution model for chloride mixed (m1CuCl2 + m2NiCl2)(aq) and (m1MgCl2 + m2CrCl3)(aq) systems is developed using the data from references [76] and [66], respectively. The ternary solubility data of Caven and Mitchel [77] for (m1(NH4)2SO4 + m2CuSO4)(aq), and

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C. Christov / J. Chem. Thermodynamics 37 (2005) 1036–1060

of Benrath and Thiemann [78] for (m1(NH4)2 SO4 + m2MgSO4)(aq) are used. Hill et al. [79] determined the pure water mssolubility of double salts (NH4)2SO4 Æ CuSO4 Æ 6H2O and (NH4)2SO4 Æ MgSO4 Æ 6H2O, which crystallize from saturated ammonium sulfate solutions (0.783 mol Æ kg1 and 0.789 mol Æ kg1, respectively). In a previous study [31], we proposed a method for evaluation of mixing parameters. This approach is applicable to ternary systems where one, or more double salts crystallize, which are congruently soluble in water at the corresponding temperature. Then, the h and w parameters can be chosen by minimizing the difference between the experimentally determined and calculated solubility of the double salt in water. The applicability of this method has been demonstrated for systems in which congruently soluble double salts with different stoichiometric compositions crystallize: 1-1-6 [31,44,64,65]; 1-1-24 [57,59]; and 2-1-2 [29,49] double salts. Therefore, the data on (NH4)2SO4 Æ CuSO4 Æ 6H2O and (NH4)2SO4 Æ MgSO4 Æ 6H2O pure water solubility [79] are also used in mixing solution model parameterization. The choice of the mixing parameters is based on the minimum deviation of the logarithm of the solubility product ðln K
sp Þ for the whole crystallization curve of the component from its value for the binary solution. In addition, the ln K
sp value for the 1-1-6 sulfate double salts crysrallizing from the saturated ternary solutions has to be constant along the whole crystallization branch of the double salt. In a previous study [27], the common chloride-, and bromide-ion ternary (m1NH4Cl + m2MgCl2)(aq), and (m1NH4Br + m2MgBr2)(aq), systems have been simulated and the theoretical solubility isotherm was plotted. The h(NH4,Mg) was set equal to (0.044) in the mixing chloride and bromide solution models. Since the parameters h(M, M 0 ) take into account only the ionic interactions of the type MM 0 in mixing solutions, their values have to be constant for the chloride, bromide, and sulfate solutions with the same cations (M+ and M 0 2+). Therefore, in constructing the mixing (m1(NH4)2SO4 + m2MgSO4)(aq), we keep the same value of h(NH4,Mg) parameter as in corresponding halide systems (equals to 0.044), i.e., only the w(NH4,Mg,SO4) have been varied. In our h and w evaluation the unsymmetrical mixing terms (Eh and Eh 0 ) have been included, according to reference [2]. The evaluated mixing parameter values are presented in table 3. The values found for ln K
sp and for the calculated pure water solubility of the double salts are given in table 4. For both ammonium sulfate double salts, model agreement with the experimental ms data are good. The solubility isotherm of the (m1CuCl2 + m2NiCl2) (aq), (m1MgCl2 + m2CrCl3)(aq), (m1(NH4)2SO4 + m2CuSO4)(aq), and (m1(NH4)2SO4 + m2MgSO4)(aq) (m1M2SO4 + m2M 0 SO4)(aq), at T = 298.15 K are calculated on the basis of the thermodynamic parameters ob-

tained. A method described in previous papers has been used [29,57,60]. At phase and chemical equilibria of a given salt (l1A1 Æ l2A2 Æ l3A3) with its saturated solution, the following equation is applied: n o ln K
sp ðl1 ; l2 ; l3 Þ ¼ l1 ln a1 þ l2 ln a2 þ l3 ln a3 ¼ constant;

ð8Þ

where a1, a2 and a3 are the activities of the components A1, A2 and A3 = H2O in the saturated solution, and l1, l2 and l3 denote the stoichiometric coefficients of the salt. Then the solubility isotherm may be calculated as a geometric locus of points meeting condition (8). The invariant point (eutonic point) of the ternary system is a point that satisfies two equations which describe the solubility isotherms of two solid phases, respectively. This eutonic represents the solution of n o ln K
sp ðl1 ; l2 ; l3 ; m1 ; m2 Þ ¼ constant0 ; n o ð9Þ ln K
sp ðl01 ; l02 ; l03 ; m1 ; m2 Þ ¼ constant00 : The predicted solubility isotherms (solid lines) are presented together with experimental data (symbols) in figures 1 to 4. Model agreement with the experimental solubility data is very good. The largest difference between model and experiment occurs for some of the experiments of Balarew and Spassow [76], which are close to the predicted (CuCl2 Æ 2H2O + NiCl2 Æ 6H2O) coexistence point in (m1CuCl2 + m2NiCl2)(aq) system (see figure 1). The model predicts lower copper chloride concentrations than experiment. In order to calculate the thermodynamic characteristics of the double salt crystallizing from the saturated ammonium sulfate ternary solutions, we have applied the scheme used successfully in references [29,31,44,56]. The theoretical basis of the adopted approach has been presented in a previous study [56]. To calculate the standard molar Gibbs free energy of reaction Dr G
m of the synthesis of the double salts from simple salts, we have used the calculated activities of the components in their saturated binary solutions (table 2). Thus, for the synthesis reaction of the ammonium-magnesium sulfate double salt (NH4)2SO4 Æ MgSO4 Æ 6H2O ðNH4 Þ2 SO4 ðcrÞ þ MgSO4  7H2 OðcrÞ ¼ ðNH4 Þ2 SO4  MgSO4  6H2 OðcrÞ þ H2 OðlÞ

ð10Þ

the change of the standard molar Gibbs free energy is n Dr G
m ¼ RT ln K
sp ððNH4 Þ2 SO4  MgSO4  6H2 OÞþ ln aðH2 OÞ  ln K
sp ððNH4 Þ2 SO4 Þ o ln K
sp ðMgSO4  7H2 OÞ ;

ð11Þ

where ln a(H2O) = 0, because the activity of pure water is 1. Using the calculated Dr G
m values and the determined in this study standard molar Gibbs free energies of for-

C. Christov / J. Chem. Thermodynamics 37 (2005) 1036–1060

1045

TABLE 3 Pitzer mixing parameters at T = 298.15 K evaluated on the basis of the ms molality in the ternary systems System

h

LiCl + MgCl2 + H2O LiBr + MgBr2 + H2O LiCl + CaCl2 + H2O LiBr + CaBr2 + H2O Li2SeO4 + NiSeO4 + H2O NaCl + NH4Cl + H2O NaBr + NH4Br + H2O Na2SO4 + (NH4)2SO4 + H2O NaBr + MgBr2 + H2O NaCl + CuCl2 + H2O NaSO4 + CuSO4 + H2O NaCl + FeCl2 + H2O Na2SO4 + FeSO4 + H2O NaCl + FeCl3 + H2O Na2SO4 + Al2(SO4)3 + H2O NaCl + CrCl3 + H2O Na2SO4 + Cr2(SO4)3 + H2O Na2SeO4 + NiSeO4 + H2O NH4Cl + MgCl2 + H2O NH4Br + MgBr2 + H2O (NH4)2SO4 + MgSO4 + H2O (NH4)2SO4 + CuSO4 + H2O (NH4)2SO4 + Al2(SO4)3 + H2O (NH4)2SO4 + Cr2(SO4)3 + H2O KCl + MgCl2 + H2O KBr + MgBr2 + H2O KCl + NiCl2 + H2O K2SO4 + NiSO4 + H2O KCl + CuCl2 + H2O K2SO4 + CuSO4 + H2O KCl + FeCl2 + H2O K2SO4 + FeSO4 + H2O KCl + FeCl3 + H2O K2SO4 + Fe2(SO4)3 + H2O KCl + AlCl3 + H2O K2SO4 + Al2(SO4)3 + H2O

0.000 0.000 0.000 0.000 0.000 0.012 0.012 0.012 0.070a 0.09 0.09 0.00 0.000 0.07 0.07 0.07 0.07 0.000 0.044 0.044 0.044 0.28 0.07 0.07 0.000a 0.000a 0.000 0.000 0.160 0.160 0.18 0.18 0.07 0.07 0.00 (I) 0.00 (II) 0.07 0.07 0.07 0.1040 0.1040 0.0 0.00 0.00 (I) 0.00 (II) 0.05 (I) 0.00 (II) 0.05 0.10 0.10 (I) 0.10 (II) 0.05 0.00 0.05 0.05 0.1260 0.1260 0.00 0.00 (I) 0.00 (II) 0.05

KCl + CrCl3 + H2O K2SO4 + Cr2(SO4)3 + H2O RbCl + MgCl2 + H2O RbBr + MgBr2 + H2O RbCl + MnCl2 + H2O RbBr + MnBr2 + H2O RbCl + CoCl2 + H2O Rb2SO4 + CoSO4 + H2O Rb2SeO4 + CoSeO4 + H2O RbCl + NiCl2 + H2O RbBr + NiBr2 + H2O Rb2SO4 + NiSO4 + H2O RbCl + CuCl2 + H2O Rb2SO4+ZnSO4 + H2O Rb2SeO4+ZnSeO4 + H2O CsCl + MgCl2 + H2O CsBr + MgBr2 + H2O CsCl + MnCl2 + H2O CsCl + CoCl2 + H2O Cs2SO4 + CoSO4 + H2O

w Common anion systems 0.003 0.001 0.007 0.011 0.015 0.0018 0.0008 0.00 0.009 0.0036 0.053 0.01 0.015 0.004 0.01 0.01 0.01 0.001 0.0180 0.0230 0.012 0.08 0.01 0.01 0.0220a 0.0265 0.031 0.069 0.001 0.060 0.003 0.03 0.01 0.03 0.06 0.08 0.01 0.01 0.01 0.0000 0.0190 0.022 0.025 0.030 0.10 0.05 0.075 0.03 0.020 0.026 0.00 0.05 0.037 0.05 0.04 0.0000 0.0367 0.00 0.00 0.09 0.04

Reference

Solubility data reference

[28] [28] [54] [54] [36] [60] [60] [60] [35] [41] [41] [65] [65] [65] [59] [62] [59] [37] [27] [27] [67] [67] [59] [59] [27] [27] [44] [44] [44] [44] [65] [65] [65] [65] [57] [57] [59] [62] [59] [27] [27] [29] [39] [29] [64] [64] [61] [64] [29] [38] [64] [64] [29] [64] [64] [27] [27] [32] [32] [64] [64]

[100] [28] [101] [54] [36] [102] [103] [104] [105] [69] [84] [106] [107] [108] [109] [62] [59,110] [37] [111] [112] [78] [77] [113] [91] [114] [112] [115] [116] [44,117] [118] [119] [120] [121] [122] [123] [124] [124] [62] [125] [27] [27] [29] [39] [29,126] [127] [127] [61] [61] [29] [38] [127] [127] [29] [128] [129] [130] [27] [131] [126] [128] [128] (continued on next page)

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C. Christov / J. Chem. Thermodynamics 37 (2005) 1036–1060

TABLE 3 (continued) System

h

w

Reference

Solubility data reference

Cs2SeO4 + CoSeO4 + H2O

(I) 0.00 (II) 0.05 0.2300 0.2300 (I) 0.23 (II) 0.05 0.05 0.05 0.000 0.0600 0.060 0.000 0.00 0.06 0.050 0.05 0.0015 0.045 0.00 0.00

0.04 0.02 0.0000 0.0199 0.015 0.13 0.05 0.08 0.050 0.0450 0.012 0.024 0.00 0.00 0.012 0.05 0.0040 0.001 0.00 0.00

[64] [33] [33] [56] [64] [64] [64] [32] [34] [65] [57] [57] [67] [65] [65] [34] [67] [60] [60]

[132] [132] [33,133] [33,134] [56] [56] [128] [129] [135] [76] [136] [123] [137] [66] [138] [122] [139] [76] [140,141] [142]

[41] [58] [58] [55] [40] [40] [55] [40]

[143,41] [58] [58] [144] [144] [145] [146,147] [148]

CsCl + NiCl2 + H2O CsBr + NiBr2 + H2O Cs2SeO4 + NiSeO4 + H2O Cs2SO4+ZnSO4 + H2O Cs2SeO4+ZnSeO4 + H2O CsCl + CuCl2 + H2O MgCl2 + CuCl2 + H2O MgCl2 + FeCl2 + H2O MgCl2 + AlCl3 + H2O MgSO4 + Al2(SO4)3 + H2O MgCl2 + CrCl3 + H2O FeCl2 + FeCl3 + H2O FeSO4 + Fe2(SO4)3 + H2O CoCl2 + CuCl2 + H2O CuCl2 + NiCl2 + H2O NiSO4 + Al2(SO4)3 + H2O CuSO4 + Al2(SO4)3 + H2O CuCl2 + CuSO4 + H2O NaCl + Na2CrO4 + H2O KCl + K2CrO4 + H2O NaCl + Na2Cr2O7 + H2O KCl + K2Cr2O7 + H2O KBr + K2Cr2O7 + H2O Na2SO4 + Na2Cr2O7 + H2O K2SO4 + K2Cr2O7 + H2O a

0.02a 0.02 0.02 0.10 0.10 0.05 0.09 0.09

Common cation systems 0.01 0.004 0.035 0.01 0.70 0.70 0.01 0.70

From Harvie et al. [7].

mation Df G
m for the components of the synthesis reaction of double salts from simple salts {reaction (10)} (table 2), we calculated the standard molar Gibbs free energy of formation Df G
m of the double (NH4)2SO4 Æ CuSO4 Æ 6H2O and (NH4)2SO4 Æ MgSO4 Æ 6H2O salts. The Dr G
m and Df G
m values obtained are given in table 4. Here, the results of the author on the thermodynamic studies at T = 298.15 K for 82 ternary (water + salt) systems where solid phases with a constant stoichiometric composition (simple and double salts) crystallize have been summarized. The values of the mixing ion-interaction parameters, which give the best fit of the experimental ternary solutions solubility data are given in table 3. The last column in table 3 gives the reference source [100–148] of the solubility data, which have been used in mixed solution parameterization. The values determined for ln K
sp ; Dr G
m ; and Df G
m of the three simple (NaBr, Na2SO4, and CaCl2 Æ 4H2O) salts and 62 double salts precipitating from the saturated mixed solutions are summarized in table 4. In the same table, the experimental and the predicted solubilities of congruently soluble double salts are given. Table 4 also compares the model calculated Df G
m values with reference data available in the literature [7,19,70,74,94,96,97]. The agreement obtained is very good. In table 4, two sets of ln K
sp ; Dr G
m and Df G
m values are presented for potassium chromium alum (K2SO4 Æ Cr2(SO4)3 Æ 24-

H2O), which correspond to the different experimental ms data of double salt solubility (0.221 mol Æ kg1 [125] and 0.408 mol Æ kg1 [149]). The data obtained on the basis of ms data of Hill et al. [149] are accepted in the model. In both calculations, we used the same mixing parameter values. With the exception of Df G
m ð2K2 SO4  Fe2 ðSO4 Þ3  14H2 OÞ [65], for all other double salts we used a calculation approach similar to those given by equations (10) and (11). Because of lack of activity data for unsaturated Fe2(SO4)3(aq) solutions, to calculate the standard molar Gibbs energy of formation of the 2K2SO4 Æ Fe2(SO4)3 Æ 14H2O double salt, we used a different scheme, proposed in reference [56]. The Df G
m ð2K2 SO4  Fe2 ðSO4 Þ3  14H2 OÞ can be determined from the following dissolution reaction: 2K2 SO4  Fe2 ðSO4 Þ3  14H2 OðcrÞ ¼ 4Kþ ðaqÞ þ 2Fe3þ ðaqÞ þ 5SO2 4 ðaqÞ þ 14H2 OðaqÞ ð12Þ by using the relation Df G
m ð2K2 SO4  Fe2 ðSO4 Þ3  14H2 OÞ ¼ RT ln K
sp ð2K2 SO4  Fe2 ðSO4 Þ3  14H2 OÞþ 4Df G
m ðKþ Þ þ 2Df G
m ðFe3þ Þ þ 5Df G
m ðSO2 4 Þþ 14Df G
m ðH2 OÞ:

ð13Þ

C. Christov / J. Chem. Thermodynamics 37 (2005) 1036–1060

1047

TABLE 4 Calculated values of the logarithm of the thermodynamic solubility product ðK
sp Þ, standard molar Gibbs free energy ðDr G
m Þ of synthesis reaction, and the standard molar Gibbs free energy ðDf G
m Þ of formation of solid phases crystallizing from saturated ternary solutions at T = 298.15 K, where ms is the molality of the saturated binary double salt-water solutionsa Salt composition

ms (exp)/(mol Æ kg1)

ms (calc)/(mol Æ kg1)

ln K
sp

Dr G
m =ðkJ  mol1 Þ

Df G
m a/(kJ Æ mol1) Calculated

NaBr Na2SO4 CaCl2 Æ 4H2O LiCl Æ MgCl2 Æ 7H2O LiCl Æ CaCl2 Æ 5H2O LiBr Æ CaBr2 Æ 5H2O Na2SO4 Æ CuSO4 Æ 2H2O Na2SO4 Æ (NH4)2SO4 Æ 4H2O Na2SO4 Æ NiSO4 Æ 4H2O Na2SO4 Æ FeSO4 Æ 4H2O Na2SO4 Æ Al2(SO4)3 Æ 24H2O Na2SO4 Æ Cr2(SO4)3 Æ 24H2O NH4Cl Æ MgCl2 Æ 6H2O NH4Br Æ MgBr2 Æ 6H2O (NH4)2SO4 Æ MgSO4 Æ 6H2O (NH4)2SO4 Æ CuSO4 Æ 6H2O (NH4)2SO4 Æ Al2(SO4)3 Æ 24H2O (NH4)2SO4 Æ Cr2(SO4)3 Æ 24H2O KCl Æ CuCl2 Æ 2H2O 2KCl Æ CuCl2 Æ 2H2O 2KCl Æ CrCl3 Æ H2O 2KCl Æ FeCl3 Æ H2O KCl Æ MgCl2 Æ 6H2O KBr Æ MgBr2 Æ 6H2O K2SO4 Æ NiSO4 Æ 6H2O K2SO4 Æ CuSO4 Æ 6H2O K2SO4 Æ FeSO4 Æ 6H2O 2K2SO4 Æ Fe2(SO4)3 Æ 14H2O K2SO4 Æ Al2(SO4)3 Æ 24H2O

K2SO4 Æ Cr2(SO4)3 Æ 24H2O RbCl Æ MgCl2 Æ 6H2O RbBr Æ MgBr2 Æ 6H2O RbBr Æ NiBr2 Æ 6H2O RbCl Æ CoCl2 Æ 2H2O Rb2SO4 Æ CoSO4 Æ 6H2O (I)

349.06 1269.98 1731.93

6.779 0.662 13.16

0.84 1.09

1.56 0.83 1.11

0.79 0.782 0.138

0.78 0.780 0.132

0.30

0.302

0.205 0.346 1.119

0.202 0.352 1.119

0.15

0.17

0.221 [125] 0.408 [149] 2.36

0.221 0.408 2.31 4.38

0.286

Rb2SeO4 Æ CoSeO4 Æ 6H2O Rb2SO4 Æ NiSO4 Æ 6H2O (I)

0.286

Rb2SO4 Æ ZnSO4 Æ 6H2O Rb2SeO4 Æ ZnSeO4 Æ 6H2O 2RbCl Æ MnCl2 Æ 2H2O 2RbBr Æ MnBr2 Æ 2H2O 2RbCl Æ CoCl2 Æ 2H2O 2RbCl Æ NiCl2 Æ 2H2O 2RbCl Æ CuCl2 Æ 2H2O CsBr Æ NiBr2 CsCl Æ CuCl2 CsCl Æ MnCl2 Æ 2H2O CsCl Æ CoCl2 Æ 2H2O CsCl Æ NiCl2 Æ 2H2O Cs2SO4 Æ CoSO4 Æ 6H2O (I)

0.236 1.805

0.286 (I) 0.286(II) 0.75 (1) 0.75 (II) 0.286 (I) 0.286(II) 0.236 0.81 1.805 3.45

2.89

2.89

0.81

0.75 (I) 0.75 (II)

21.70 24.4 34.62 9.02 3.86 7.87 7.4 19.9 18.76 9.590 12.000 10.23 11.6 28.25

2.01 7.51 24.71 0.45 3.97 0.0 1.34 7.33 9.21 9.090 8.081 16.14 15.17 36.46

2748.45 2365.40 2257.20 2443.37 3128.71 3024.95 3058.63 10221.01 9669.68 2326.60 2238.98 3552.18 3033.71 9877.26

26.12 5.12 6.44 19.0 12.97 10.109 12.760 14.33 13.26 10.61 30.04 27.4

35.89 3.57 5.45 3.18 5.502 5.85 5.106 12.84 7.86 3.46 22.85

9323.47 1068.61 1479.63 1628.18 1432.35 2529.63 2441.47 3558.91 3446.10 3583.58 8259.40 10283.35

27.32 21.66 15.720 10.361 6.15 0.00 15.40

9734.60 9728.7 2538.16 2447.85 2034.44 1184.28 3566.83

18.34

2953.66

16.12

3558.87

19.04 17.53 13.88 7.24 3.79 2.38 20.25 7.75 14.52 8.53 2.28 5.21 18.49

3661.26 3044.45 1778.82 1654.08 1596.08 1582.65 1491.75 607.39 610.70 1372.40 1193.49 1184.41 3576.38

27.3 25.02 7.100 10.620 12.14 8.96 13.13 (I) 13.196 (II) 10.46 (I) 10.46 (II) 13.537 (I) 13.47 (II) 13.88 9.7 6.90 13.18 10.40 12.20 2.40 17.05 2.10 6.45 8.40 8.25 10.544 (I) 10.513 (II)

Reference data 1270.02 [7] 1731.95 [7] 1729.43 [19]

3129.7 [74]

9876.51 [94] 9875.99 [97]

2529.14 [7]

10284.2 [94] 10283.77 [97] 10281.5 [96]

(continued on next page)

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C. Christov / J. Chem. Thermodynamics 37 (2005) 1036–1060

TABLE 4 (continued) ms (exp)/ (mol Æ kg1)

Salt composition

Cs2SeO4 Æ CoSeO4 Æ 6H2O C2SO4 Æ NiSO4 Æ 6H2O (I) Cs2SeO4 Æ NiSeO4 Æ 6H2O

a

0.738 3.25

Dr G
m =ðkJ  mol1 Þ

ln K
sp

Df G
m a/(kJ Æ mol1) Calculated

8.86 (I) 9.01 (II) 12.04 (I) 11.976 (II) 9.40 (I) 9.38(II) 10.78 9.26 10.400 12.660 7.60 10.65 13.80 2.40 3.65 5.43 19.77 18.01

1.1 (1) 1.1 (II) 0.6 (I) 0.495 (II) 0.83 (I) 0.83 (II) 0.738 1.16 3.26

0.495

Cs2SO4 Æ ZnSO4 Æ 6H2O Cs2SeO4 Æ ZnSeO4 Æ 6H2O CsCl Æ MgCl2 Æ 6H2O CsBr Æ MgBr2 Æ 6H2O 2CsCl Æ MnCl2 Æ 2H2O 2CsCl Æ CoCl2 3CsCl Æ CoCl2 2CsCl Æ CuCl2 3CsCl Æ 2CuCl2 Æ 2H2O 4CsCl Æ 3CuCl2 Æ 2H2O MgCl2 Æ FeCl2 Æ 8H2O MgSO4 Æ Al2(SO4)3 Æ 22H2O

ms (calc)/(mol Æ kg1)

15.70

2965.96

22.03

3571.25

19.01

2958.3

20.97 17.575 8.668 3.631 14.33 5.35 6.20 22.43 39.07 54.38 5.528 0.58

3669.66 3059.43 2537.84 2450.74 1792.73 1136.84 1552.2 1507.3 2120.20 2731.69 2906.63 9675.8

Reference data

9674.9 [70]

s

The parenthesized values in the m (exp) column are references.

In references [29,32], a mistake has been made during the calculation of Df G
m for the double mMCl Æ nM 0 Cl2 Æ xH2O (M = Rb or Cs; M 0 = Mn, Co, Ni, Cu; m = 1, 2, 3, or 4; n = 1, 2, or 3; x = 0 or 2) salts, for which the author should like to apologize. Here, the reader will find the correct Df G
m data for these double salts (table 4). Note that with the exception of the osmotic data for Na-Cu-Cl-SO4-H2O {Downes and Pitzer [81]}, (m1NaCl + m2NH4Cl)(aq) [150,151], and (m1Na2SO4 + m2(NH4)2SO4)(aq) [152] systems, for all other mixing solutions considered here, measurements of

aqueous mixtures activities (activity coefficients, osmotic coefficients, water activity, e.m.f. data) are not available. Hence, the mixing ion interaction parameters (h and w) for the ternary systems in which stoichiometric compounds crystallize, were evaluated using only the ternary solutions solubility data. For systems in which congruently soluble double salts crystallize, the minimum deviation of the calculated pure water solubility from those experimentally obtained was an important criterion in h and w evaluations. Note that the solubility data in multicomponent solutions are not included in the mixing parameter evaluations.

6

5 NiCl2.6H2O

4 CrCl3.6H2O

4

m 2/(mol.kg-1)

m2/(mol.kg-1)

5

CuCl2.2H2O

3 2

3

2 MgCl2.6H2O

1

1 0 0

1

2

3 4 m1/(mol.kg-1)

5

6

7

FIGURE 1. Molality m (sat) solubilities in (m1CuCl2 + m2NiCl2)(aq) at T = 298.15 K with m2 plotted against m1: ——, model predictions; symbols: experimental data of Balarew and Spassow [76] (h: CuCl2 Æ 2H2O(cr) solubility; n: NiCl26H2O(cr) solubility).

0 0

1

2

3 4 -1 m1/(mol.kg )

5

6

7

FIGURE 2. Molality m(sat) solubilities in (m1MgCl2 + m2CrCl3)(aq) at T = 298.15 K with m2 plotted against m1: —— predicted values; h, n: experimental [66] CrCl36H2O(cr) and MgCl26H2O(cr) solubility, respectively.

C. Christov / J. Chem. Thermodynamics 37 (2005) 1036–1060

2

CuSO4.5H2O

m 2/(mol.kg-1)

1.5

(NH4)2SO4.CuSO4.6H2O

1

0.5 (NH4)2SO4

0

0

1

2

3 4 -1 m 1/(mol.kg )

5

6

7

FIGURE 3. Molality m(sat) solubilities in (m1(NH4)2SO4 + m2CuSO4)(aq) at T = 298.15 K with m2 plotted against m1: ——, predicted values; n, s, and }, symbols represent the experimental CuSO4 Æ 5H2O(cr), (NH4)2SO4 Æ CuSO4 Æ 6H2O(cr) and (NH4)2SO4(cr) solubility data, respectively {Caven and Mitchell [77]}. The closed circle shows the experimental pure water solubility of the (NH4)2SO4 Æ CuSO4 Æ 6H2O double salt [79].

3.5

MgSO4.7H2O

3

m 2/(mol.kg-1)

2.5 2

(NH4)2SO4.MgSO4.6H2O

1.5 1 (NH4)2SO4

0.5 0 0

1

2

3 4 -1 m1/(mol.kg )

5

6

7

FIGURE 4. Molality m(sat) solubilities in (m1(NH4)2SO4 + m2MgSO4)(aq) at T = 298.15 K with m2plotted against m1: ——, predicted values; n, s, } and, symbols represent the experimental MgSO4 Æ 7H2O(cr), (NH4)2SO4 Æ MgSO4 Æ 6H2O(cr) and (NH4)2SO4(cr) solubility data, respectively {Benrath and Thiemann [78]}. The closed circle shows the experimental pure water solubility of the (NH4)2SO4 Æ MgSO4 Æ 6H2O double salt [79].

However, when data in more complex systems are available, these data are used in the test of model parameterization (see next section). In parameterization of (m1NaBr + m2MgBr2)(aq) [35], (m1KBr + m2MgBr2)(aq) [27], and (m1CuCl2 + m2-

1049

CuSO4)(aq) [41] solutions, we have varied the w values only, assuming h(Na,Mg) = 0.07, h(K,Mg) = 0.00, and h(Cl,SO4) = 0.02 from the model of Harvie et al. [7] at T = 298.15 K. Note also, that (m1KCl + m2MgCl2)(aq) mixing model of Harvie et al. [7] was re-parameterized (reference [27]), because we used new MgCl2(aq) single electrolyte parameters (table 1). It was established that a mixing model with new ln K
sp ðKCl  MgCl2  6H2 O ðcarnalliteÞÞ value (table 4), and the same h(K,Mg) and w(K,Mg,Cl) mixing parameter values (table 3) {as in Harvie et al. [7]} fit very well the experimental ternary solubility data. The solutions containing the same cations [e.g., (m1LiCl + m2MgCl2)(aq) and (m1LiBr + m2MgBr2)(aq)] or the same anions [e.g., (m1NaCl + m2Na2CrO4)(aq) and (m1KCl + m2K2CrO4)(aq)] have been parameterized simultaneously in order to preserve the same value of the binary mixing h parameter. Note that in the case of common ion systems, the simulation of ternary aqueous mixtures began with the system, about which the solubility data are more complete [64]. The thermodynamic data presented in tables 3 and 4 permit the following most general correlations, which facilitate the mixing solution parameterization process. The same values for both h and w parameters, or for the h binary mixing parameter only, describe correctly the properties of the ternary systems with similar types of phase diagram (similar solid phases precipitate from saturated ternary solutions). It was shown that zero values of both mixing interaction parameters (h = w = 0.0) fit very well the experimental solubility data of ternary aqueous mixtures of 2-2 and 3-2 electrolytes (m1MSO4 + m2Al2(SO4)3)(aq) (M = Mg, Cu, Ni) [57,60]. In constructing the model for ternary aqueous mixtures of 1-2 and 3-2 electrolytes ðm1 M2 SO4 þ m2 M02 ðSO4 Þ3 ÞðaqÞ (M = NH4, Na, K; M 0 = Al, Cr), we accepted h = 0.07 and w = 0.01 [59]. The model for (m1M2SO4 + m2M 0 SO4)(aq) and (m1M2SeO4 + m2M 0 SeO4)(aq) (M = Rb, Cs; M 0 = Co, Ni, Zn) [64] systems used the same value for binary mixing parameter (h = 0.05). Note that the mixing solution model for ternary (m1M2SO4 + m2M 0 SO4)(aq), (m1M2SeO4 + m2M 0 SeO4)(aq) (M = Rb, Cs; M 0 = Co, Ni, Zn) [56,61,64], and (m1K2SO4 + m2Al2(SO4)3)(aq) [57,59] systems have been constructed using two different approaches [(I) and (II) in tables 3 and 4]. During the first stage of our thermodynamic analysis [approach (I)], we simulated the solubilities of the above ternary mixtures keeping the same values of h(M, M 0 ) parameters as in corresponding halide systems, i.e., only the w(M,M 0 ,SO4), w(M,M 0 ,SeO4), and w(K,Al,SO4) have been varied [56,57,61]. To validate the mixing solutions model for these systems, for which the experimental data are quite limited, in approach (II) we accept the same h(M, M 0 ) value for all (i) (m1M2SO4 + m2M 0 SO4)(aq), and (m1M2SeO4 + m2MeSeO4)(aq) systems [64], and

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C. Christov / J. Chem. Thermodynamics 37 (2005) 1036–1060

for all six (ii) ðm1 M2 SO4 þ m2 M02 ðSO4 Þ3 ÞðaqÞ (M = NH4, Na, K; M 0 = Al, Cr) systems [59]. The agreement between the ln K
sp and ms values for double salts crystallizing in above ternary systems, calculated using the different approach in evaluation of mixing parameters is very good (table 4). The largest K
sp difference is 2% (for Cs2SeO4 Æ CoSeO4 Æ 6H2O double salt) or less (table 4). Note that, for double salts, for which the application of approach I and II in mixing solution parameterization give a different K
sp values (table 4), an average of the two values have been used in the Dr G
m determination. According to Pabalan and Pitzer [13], any negative values of mixing parameters tend to increase the solubility in ternary systems, while positive values have the opposite effect. According to previous studies of the author [44,59,64,65,67], the effect of the sign of the ternary solution parameters is the same. However, the value of the binary mixing parameter h determines the type of the calculated solubility diagram at low concentration of the second component, while the sum of both mixing parameters (h + w) determines the predictions at high concentration of both components. Following the last suggestion, for systems in which the ternary solubility of some component increase (compare to the binary solubility) with an increase in concentration of second component, binary mixing h(M, M 0 ) parameter should have a negative value. Therefore, in the model for (m1M2SO4 + m2M 0 SO4)(aq), (m1M2SeO4 + m2M 0 SeO4) (aq) (M = Rb or Cs; M 0 = Co, Ni, Zn) [64], ðm1 M2 SO4 þ m2 M 02 ðSO4 Þ3 ÞðaqÞ (M = Na, K, NH4; M 0 = Al, Cr) [59], and (m1M2SO4 + m2CuSO4)(aq) (M = NH4, K) [44,67] systems binary mixing h(M, M 0 ) parameter have a negative value (see table 3). The system (m1K2SO4 + m2Cr2(SO4)3)(aq) was studied [59] by evaluating at first the optimum values of the binary parameters for the solution KCr(SO4)2(aq). On the basis of the pure electrolyte parameters presented in table 1 and the ms molality of the binary solution, we calculated ln K
sp for potassium chromium alum K2SO4 Æ Cr2(SO4)3 Æ 24H2O (table 4). The minimum deviation of ln K
sp for the double salt in the mixing (m1K2SO4 + m2Cr2(SO4)3)(aq) solution from its value in the saturated binary solution was an important criterion for the choice of mixing parameters. In the case of double salts with the stoichiometric composition 1-1-n, where n is the number of molecules of crystal water, the Dr G
m values are directly dependent on n: the larger n, the more favourable thermodynamically the formation of the double salt from the corresponding simple salts which are components of the ternary system (table 4). For that reason, the crystallization of the double salts 1-1-2 is most unfavourable thermodynamically [29,32,41], whereas the crystallization of the alums and chromium alums is accompanied by the largest change of Dr G
m . [59].

Proceeding from the crystal structure of M2SO4 Æ M 0 SO4 Æ 6H2O and M2SeO4 Æ M 0 SeO4 Æ 6H2O (M = Rb, Cs; M 0 = Ni, Co, Zn) double salts, it was concluded that the high ionic radii of the cations {r(M+) and r(M 0 2+)}, and of sulfate anions are the favourable factors that determine crystallization and stability of double salts [64]. The Df G
m values of 1-1-6 sulfate and selenate double salts presented in table 4 confirm this conclusion. According to Df G
m data, under standard conditions, the crystallization of cesium-zinc-sulfate 11-6 double salt is thermodynamically more favourable (more negative Df G
m values) than the crystallization of other sulfate, and selenate 1-1-6 double salts. In studies on carnallite type solutions (m1MX + m2MgX2)(aq) (M = Li, K, NH4, Rb, Cs; X = Cl, Br) [27,28,31,153,154], and those of the type (m1MX + m2M 0 X2)(aq), where X denotes Cl or Br, M stand for Rb or Cs, and M 0 for Mn, Co, Ni, or Cu [29,32,34,44] we have demonstrated some correlations between thermodynamic characteristics of the double salts crystallizing from the saturated ternary solutions and the type of the solubility isotherms. Generally speaking, low Dr G
m and Df G
m values for a given double salt correspond to a low solubility and a broad crystallization field, and vice versa. For chloride and bromide systems [27,28,34] containing the same cations (M+ and M 0 2+), where double salts with the same stoichiometric composition crystallize, the following relationships have been found: (i) the absolute value of the sum of the ternary parameters for the bromide systems is larger than that for the corresponding chloride system and (ii) the Dr G
m values for the chloride double salts are lower than those for the corresponding bromide double salts. Proceeding from the foregoing and taking into account the condition about thermodynamic stability of double salts (a negative Dr G
m value), we have constructed a preliminary mixing solution model, and plotted preliminary theoretical solubility isotherms for the (m1RbBr + m2NiBr2)(aq) [38] and (m1RbBr + m2MnBr2)(aq) [39] systems. The experiments have confirmed the results of the calculation. For most of the systems investigated, the model predicted solubilities in ternary mixtures agree very well with the experimental data obtained by us [27–29,33,36–38,54,58,61,62] and those given in the literature [100–148]. Additional experiments were performed [27–29,31,41,44] to check contradictory data reported in the literature and specify the type of the solubility diagrams. These experiments confirmed the predicted compositions of the saturated ternary solutions. 3.1.3. Quaternary systems: validation of model parameters The model, presented in tables 1 to 4, is parameterized from the data in binary and common-ion ternary

C. Christov / J. Chem. Thermodynamics 37 (2005) 1036–1060

systems. Therefore, when data are available in more complex mixed aqueous solutions (not used in the model parameterization) comparisons are made to these data. The calculated thermodynamic functions [binary (table 1) and mixing (table 3) ion-interaction parameters and thermodynamic solubility product of solids (table 2 and 4)] are used in this test. In previous studies, the results of the model tests for {Na + Cu + Cl + SO4 + H2O} reciprocal system [41], and in the {Na + NH4 + Cr + SO4 + H2O} [60], {Na + NH4 + Al + SO4 + H2O} [63], {Na + Ni + Al + SO4 + H2O} [60], {Na + Cl + SO4 + Cr2O7 + H2O} [55], {K + Mg + Al + Cl + H2O} [57], {K + Mg + Fe + Cl + H2O} [65], and {K + Cl + SO4 + Cr2O7 + H2O} [40] quaternary systems are presented. Note that only stoichiometric solids (simple and double salts) crystallize from the above multicomponent solutions. In the model test, the mixing solution model for (m1NaCl + m2Na2SO4)(aq) and (m1KCl + m2K2SO4)(aq) subsystems are taken from Harvie et al. [7]. The composition of the quaternary invariant points was calculated as points where the chemical potential of all three solid phases is constant. The results of the calculation are presented as a horizontal non-aqueous projection of the solubility diagram, presented as a triangular [40,55,57,60], or quadrangular (for reciprocal systems) [41] prism. Here, we summarized the results of model tests in table 5. The calculated (T = 298.15 K) composition (in mol Æ kg1) of invariant points in the quaternary systems are compared with experimental data [113,119,123,141,147,155–157]. The chloride concentration in the {NaCl + CuCl2 + Na2SO4 + CuSO4 + H2O} system can be obtained from the charge balance. Note that the composition of three salts saturated solutions, calculated at T = 298.15 K for systems {NaCl + Na2SO4 + Na2Cr2O7 + H2O} and {Na2SO4 + (NH4)2SO4 + Cr2(SO4)3 + H2O} have been compared with those obtained experimentally at T = 293.15 K [155,156]. For K-Mg-Fe(II)-Cl-H2O quaternary the data are at T = 295.95 K [119]. The type of the solubility diagrams of the ternary systems, which are a subsystems for above quaternary, at T = (293.15, 295.95, 298.15) K is the same, i.e., the same solid phases crystallize in them. This fact has permitted the above comparison. With three exceptions the model is in very good agreement with experiment. For (Na2SO4 Æ 10H2O + Na2SO4 Æ (NH4)2SO4 Æ 4H2O + (NH4)2SO4 Æ Cr2 (SO4)3 Æ 24H2O) invariant point in the {Na2SO4 + (NH4)2SO4 + Cr2(SO4)3 + H2O} quaternary, the model predicts higher sodium sulfate and lower ammonium sulfate concentrations than the data [156]. The model predicts higher magnesium and lower potassium and Fe(II) concentration for three salts (KCl + KCl Æ MgCl2 Æ 6H2O + FeCl2 Æ 4H2O) co-existence point in {K + Mg + Fe(II) + Cl + H2O} system, than the data [119]. For the solution saturated with (KCl + KCl Æ MgCl2 Æ 6H2O + AlCl3 Æ 6H2O) solid phases in

1051

{K + Mg + Al + Cl + H2O} system, the model predicts higher magnesium and lower aluminum concentrations, compare to the experimental data [123]. Probable reasons for these differences are: (1) the effect of temperature on the width of the equilibrium crystallization fields in corresponding ternary subsystems (for first two systems); (2) experimental uncertainty; (3) the ln K
sp value for carnallite used in this test ðln K
sp ¼ 10:109Þ (for chloride systems). This value was determined from solubility data in ternary (m1KCl + m2MgCl2)(aq) at T = 298.15 and gives an excellent agreement with experiment [27]. Note that the predicted composition of invariant points, as well as the predicted phase diagrams, are model dependent. To reproduce the results given in table 5, it is necessary to use the same values for all thermodynamic functions (pure electrolyte parameters, mixing parameters, and ln K
sp of solids) as in the model, presented in tables 1 to 4. To illustrate the last statement, we re-simulate the solubilities in {NaCl + CuCl2 + Na2SO4 + CuSO4 + H2O} reciprocal system, using different sets of pure electrolyte parameters for sodium solutions. In both calculations [(I) and (II) in table 5], the binary parameters for CuCl2(aq) and CuSO4(aq), given in table 1, have been used along with the values of the mixing parameters from table 3 and those of ln K
sp for simple and double salts given in tables 2 and 4. In variant (I) {presented in reference [41]}, the pure electrolyte parameters of Filippov et al. [69] for NaCl(aq), and those of Kim and Frederick [80] for Na2SO4(aq) have been used. Here, we have repeated the quaternary system solubility calculations [variant (II) in table 5], exchanging only the single electrolyte parameters for Na2SO4(aq) and NaCl(aq), which have been taken from Harvie and Weare [6]. As is shown in table 5, the new model parameters deteriorate considerably the agreement with the experimental data, especially at high sodium concentration, where Na2SO4(aq) and NaCl(aq) binary parameters become very important. The main reason for this deviation is that the mixing h(Na,Cu), w(Na,Cu,Cl) and w(Na,Cu,SO4) (table 3) have been evaluated using different sets of pure electrolyte parameters. In previous investigations [42,43], we proposed an approach for solubility calculations in multicomponent systems in which both phases, with a constant composition, and solid solutions crystallize. The phase diagram of carnallite type quaternary systems {RbCl + CsCl + MgCl2 + H2O}, {RbCl + KCl + MgCl2 + H2O} and {RbCl + RbBr + MgCl2 + MgBr2 + H2O} were plotted at T = 298.15 K. The mixing parameters of the subsystems in which phases with a varying composition {(m1RbCl + m2CsCl)(aq), (m1RbCl + m2KCl)(aq), (m2RbCl + m2RbBr)(aq), and (m1MgCl2 + m2MgBr2) (aq)} crystallize, were determined using the rule of Zdanovskii [158]. As is known, many of the (water + salt)

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C. Christov / J. Chem. Thermodynamics 37 (2005) 1036–1060

TABLE 5 Experimental at T = (293. 15, 295.95, and 298.15) K and model predicted at T = 298.15 K compositions (units: mol Æ kg1) of invariant points for the quaternary systems mNaCl Calculated Experimentala

Calculated Experimentalb Calculated Experimentalb Calculated Experimentalb Calculated Experimentalb

Calculated Experimentalc Calculated Experimentalc Calculated Experimentalc

Calculated Experimentald Calculated Experimentald

Calculated Experimentale

Calculated(I)f Calculated(II)g Experimentalh Calculated(I)f Calculated(II)g Experimentalh Calculated(I)f Calculated(II)g Experimentalh Calculated(I)f Calculated(II)g Experimentalh

Calculated Experimentali Calculated Experimentali

mNa2 SO4

mNa2 Cr2 O7

0.84 0.79

0.09 0.05

mNa2 SO4

mðNH4 Þ2 SO4

0.90 0.93 2.76 2.21 2.17 2.13 0.35 0.43

5.35 5.40 1.60 2.19 0.06 0.06 0.01 0.03

mNa2 SO4

mNiSO4

2.00 2.04 0.27 0.10 0.83 0.90

0.60 0.50 1.22 1.24 1.76 1.56

mKCl

mMgCl2

Solid phases

NaCl + Na2SO4 + Na2Cr2O7 + H2O system (reference [55]) 6.57 NaCl + Na2SO4 + Na2Cr2O7 6.62 mCr2 ðSO4 Þ3

Solid phases

Na2SO4 + (NH4)2SO4 + Cr2(SO4)3 + H2O system (reference [60]) 0.12 (NH4)2SO4 + Na2SO4 Æ (NH4)2SO4 Æ 4H2O + (NH4)2SO4 Æ Cr2(SO4)3 Æ 24H2O 0.13 0.19 Na2SO4 Æ 10H2O + Na2SO4(NH4)2SO4 Æ 4H2O + (NH4)2SO4 Æ Cr2(SO4)3 Æ 24H2O 0.10 0.87 Na2SO4 Æ 10H2O + Na2SO4Cr2(SO4)3 Æ 24H2O + (NH4)2SO4 Æ Cr2(SO4)3 Æ 24H2O 0.96 1.60 Cr2(SO4)3 Æ 16H2O + Na2SO4Cr2(SO4)3 Æ 24H2O + (NH4)2SO4 Æ Cr2(SO4)3 Æ 24H2O 1.38 mAl2 ðSO4 Þ3

Solid phases

Na2SO4 + NiSO4 + Al2(SO4)3 + H2O system (reference [60]) 0.42 NaSO4 Æ 10H2O + Na2SO4NiSO4 Æ 4H2O + Na2SO4 Æ Al2(SO4)3 Æ 24H2O 0.436 0.81 NiSO4 Æ 7H2O + Al2(SO4)3 Æ 17H2O + Na2SO4 Æ Al2(SO4)3 Æ 24H2O 0.82 0.43 NiSO4 Æ 7H2O + Na2SO4NiSO4 Æ 4H2O + Na2SO4 Æ Al2(SO4)3 Æ 24H2O 0.46 mAlCl3

Solid phases

KCl + MgCl2 + AlCl3 + H2O system (reference [57]) 2.80 KCl + AlCl3 Æ 6H2O + KCl Æ MgCl2 Æ 6H2O 3.24 0.85 MgCl2 Æ 6H2O + AlCl3 Æ 6H2O + KCl Æ MgCl2 Æ 6H2O 0.59

0.46 0.42 0.01 0.02

1.04 0.59 4.87 4.89

mKCl

mK2 SO4

4.62 4.69

0.07 0.12

mNa

mCu

1.65 1.52 1.52 3.41 3.14 3.49 6.67 6.38 6.85 6.34 5.92 6.32

NaCl + CuCl2 + Na2SO4 + CuSO4 + H2O system {References [41,67]} 5.47 0.33 CuCl2 Æ 2H2O + CuSO4 Æ 5H2O + Na2SO4 Æ CuSO4 Æ 2H2O 5.40 0.32 5.57 0.36 5.27 0.10 CuCl2 Æ 2H2O + NaCl + Na2SO4 Æ CuSO4 Æ 2H2O 5.05 0.09 5.17 0.17 0.34 0.72 NaCl + Na2SO4 + Na2SO4 Æ CuSO4 Æ 2H2O 0.33 0.60 0.22 0.75 0.19 1.56 Na2SO4 + Na2SO4 Æ 10H2O + Na2SO4 Æ CuSO4 Æ 2H2O 0.19 1.28 0.11 1.60

mKCl

mMgCl2

0.79 1.30 0.02 0.06

2.88 2.25 5.44 5.39

mK2 Cr2 O7

Solid phases

KCl + K2SO4 + K2Cr2O7 + H2O system (reference [40]) 0.03 KCl + K2SO4 + K2Cr2O7 0.05 mSO4

mFeCl2

Solid phases

Solid phases

KCl + MgCl2 + FeCl2 + H2O system (reference [65]) 2.50 KCl + FeCl2 Æ 4H2O + KCl Æ MgCl2 Æ 6H2O 3.11 0.79 MgCl2 Æ 6H2O + FeCl2 Æ 4H2O + KCl Æ MgCl2 Æ 6H2O 0.78

C. Christov / J. Chem. Thermodynamics 37 (2005) 1036–1060

1053

TABLE 5 (continued) mNa Calculated Experimentalj a b c d e f g h i j

0.82 0.909

mAl

mNH4

Solid phases

Na2SO4 + (NH4)2SO4 + Al2(SO4)3+H2O system (reference [63]) 0.01 0.83 Na2SO4 Æ Al2(SO4)3 Æ 24H2O + (NH4)2SO4 Æ Al2(SO4)3 Æ 24H2O 0.003 0.912

Data of Zhukov and Shutova [155] at T = 293.15 K. Data at T = 293.15 K from reference [156]. From reference [141]. From reference [123]. From reference [147]. Model predictions (I) from reference [41]. This study (II) [67]. From reference [157]. Data at T = 295.95 K from reference [119]. From reference [113].

systems in which solid solutions are formed, obey the rule of Zdanovskii, i.e., the isoactivities of water fall on straight lines over the whole concentration range of the solutions [158]. Assuming the applicability of the rule of Zdanovskii to the above systems, we have calculated the mixing parameters on the basis of data for the binary subsystems only. The values found are presented in table 9. The solubilities in the above ternary subsystems, in which mixed crystals precipitate, are calculated, assuming that only stoichiometric simple salt, components of the system, crystallize in them (reference [43]). New experimental quaternary solubility data are required to validate these approximations and model predictions.

[159] for estimation of osmotic coefficients. The authors found that over a wide temperature range the difference between the osmotic coefficients of a given electrolyte and another similar standard electrolyte, (/  /st), can be approximated by a linear relationship. The authors calculated the osmotic coefficients at a temperature T according to the following equation:

3.1.4. Thermodynamic model at temperatures differing from the standard In previous studies by the author, the results from thermodynamic and experimental solubility investigations on the systems (m1NaBr + m2MgBr2)(aq) at T = 273.15 K (reference [35]) and (m1LiBr + m2MgBr2)(aq) at T = 348.15 K (reference [30]) are presented. The mixing solutions equilibrium model was constructed as described in detail previously (section 3.1.2). The main problem in constructing the mixing model was the determination of pure electrolyte ion interaction parameters for binary subsystems at T = (273.15 and 348.15) K. The lack of initial activity data at these temperatures did not allow direct evaluation of b(0), b(1), and Cu parameters. These parameters were calculated using the approach of Apelblat et al.

Here, y+ and y denote the number of cations and anions to which the electrolyte is dissociated, y = y+ + y. The numerical coefficient B equals to 0.08634 for 1-1 type electrolytes, while for 1-2 and 21 electrolytes B was set equal to 0.01437. The osmotic coefficients /(LiBr), /(NaBr) and /(MgBr2) at T = (273.15 and 348.15) K were calculated according to equations (14) and (15), the standard electrolytes chosen being LiCl, NaCl and MgCl2. The binary parameters for NaCl(aq) at T = 273.15 K were taken from Pitzer et al. [160]. The binary parameters for MgCl2(aq) at T = 273.15 K, LiCl(aq) at T = 348.15 K, and MgCl2(aq) at T = 348.15 K were calculated using the empirical approximating equation proposed by Spencer et al. [9], Holmes and Mesmer [161], and de Lima and Pitzer [162], respectively. The pure

/ðtÞ ¼ /st ðtÞ þ kðtÞm;

ð14Þ

where k(t) is

nh io kðtÞ ¼ 2ðy þ y  =yÞ bð0Þ ð298:15 KÞ  bstð0Þ ðtÞ þ nh io B bð1Þ ð298:15 KÞ  bstð1Þ ðtÞ :

ð15Þ

TABLE 6 Assessed Pitzer pure electrolyte parameters at T = (273.15 and 348.15) K for the LiBr(aq), NaBr(aq), and MgBr2(aq) solutions System b

NaBr + H2O MgBr2 + H2Ob LiBr + H2Oc MgBr2 + H2Oc a b c

T/K

k(t)a

b(0)

b(1)

Cu

mmax/(mol Æ kg1)

r

273.15 273.15 348.15 348.15

0.0508 0.0962 0.0268 0.1416

0.10016 0.43849 0.16751 0.47369

0.24611 1.31484 0.34017 0.65919

0.00516 0.00938 0.00164 0.01260

8.00 5.50 23.7 6.25

0.00006 0.00059 0.00006 0.00270

k(t) is defined in equation (15). From reference [35]. From reference [30].

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C. Christov / J. Chem. Thermodynamics 37 (2005) 1036–1060

TABLE 7 Calculated values of the logarithm of the thermodynamic solubility product, K
sp , of solid phases at T = (273.15 and 348.15) K Salt composition

T/K

ms/(mol Æ kg1)

ln K
sp

NaBr Æ 2H2Oa MgBr2 Æ 6H2Oa LiBr Æ H2Ob MgBr2 Æ 6H2Ob

273.15 273.15 348.15 348.15

7.82 5.22 23.7 6.25

4.976 12.464 17.45 14.71

a b

From reference [35]. From reference [30].

electrolyte parameter and k(T) values obtained are given in table 6, while the ln K
sp values of crystalline hydrates are presented in table 7. The mixing parameters for (m1NaBr + m2MgBr2)(aq) at T = 273.15 K (reference [35]) and (m1LiBr + m2MgBr2)(aq) at T = 348.15 K [30] are evaluated from the experimental ternary solution solubilities given by Fedunyak and Bogoyavlenskii [105] and obtained by us [30], respectively. By analogy with Pabalan and Pitzer [12] and Spenser et al. [9] on the effect of temperature on the values of the mixing parameters, we assumed that h(Li,Mg) and h(Na,Mg) are temperature independent and remain the same values as at T = 298.15 K, i.e., h(Li,Mg) = 0.0 and h(Na,Mg) = 0.07 (see table 3). We varied the values of w(Li,Mg,Br) and w(Na,Mg,Br) only. The best agreement with the experiment was obtained with w(Li,Mg,Br) =  0.0004 (348.15 K) [30], and w(Na,Mg,Br) = 0.010 (273.15 K) [35]. 3.2. Calculation of the molar Gibbs free energy of mixing of crystals McCoy and Wallace [163] calculated the standard molar Gibbs free energy of mixing Dmix G
m ðsÞ of K(Cl,Br) crystals according to the following equation: Dmix G
m ðsÞ=RT ¼ x1 ln a1 ðsÞ þ x2 ln a2 ðsÞ ¼ x1 fln a1 ðlÞ  ln a1 ðlo Þgþ x2 fln a2 ðlÞ  ln a2 ðlo Þg;

ð16Þ

where xi is the mole fraction of component i in the mixed crystals, ai is the activity, and the indices s, l, and lo refer to mixed crystals, and saturated ternary and binary solutions, respectively. Filippov and Rumianzev [22] have used equation (16) in combination with Pitzer ion-interaction model for systems where mixed crystals are formed. In previous studies, we demonstrated the applicability of equation (16) for Dmix G
m ðsÞ calculations for a series of systems in which isomorphous or isodimorphous mixed crystals of both anhydrous salts [45,46,48], and crystalline hydrates [43–45,47,49– 51,63,65] are formed. The excess Gibbs free energy of mixing Dmix GEm ðsÞ was calculated using the following equation: Dmix GEm ðsÞ ¼ Dmix G
m ðsÞ  Dmix Gid m ðsÞ:

ð17Þ

The extrapolated dependence of Dmix G
m ðsÞ and Dmix GEm ðsÞ on the mole fraction of one of the components [plots of Dmix G
m ðsÞ against xi, and Dmix GEm ðsÞ against xi illustrate that the Dmix G
m ðsÞ and Dmix GEm ðsÞ values obtained can be described by a dependence which is very close to being symmetric and has a minimum at xi  0.5 [43–49,65]. This permits the conclusion that the regular solution model describes sufficiently well the mixed crystals precipitating from saturated solutions. Using the regular mixing model the Dmix G
m ðsÞ and Dmix GEm ðsÞ results obtained may be summarized by the following simplified equations, which give the dependence of Dmix G
m ðsÞ and Dmix GEm ðsÞ on the composition of the mixed crystals Dmix G
m ðsÞ=ðkJ  mol1 Þ ¼ Ax1 x2 ;

ð18Þ

Dmix GEm ðsÞ=ðkJ  mol1 Þ ¼ GE1 ðsÞx1 x2 :

ð19Þ

In reference [49], equation (16) has been used for determination of Dmix G
m ðsÞ for mixed crystals of 2RbCl Æ NiCl2 Æ 2H2O and 2RbCl Æ MnCl2 Æ 2H2O double salts.The scheme proposed has been applied to thermodynamic simulation of (m1K2SO4NiSO4 + m2K2SO4CuSO4)(aq) {K2SO4 Æ (Ni,Cu)SO4 Æ 6H2O mixed crystals} [44], [m1(NH4)2SO4 Æ Al2(SO4)3 + m2K2SO4Al2(SO4)3 Æ 24H2O](aq) {(NH4,K)2SO4 Æ Al2(SO4)3 Æ 24H2O mixed crystals} [63], [m1(NH4)2SO4 Æ Al2(SO4)3 + m2(NH4)2SO4 Æ Cr2(SO4)3](aq) {(NH4)2SO4 Æ (Al,Cr)2(SO4)3 Æ 24H2O mixed crystals} [63], and [m1(NH4)2SO4 Æ Cr2(SO4)3 + m2K2SO4 Æ Cr2(SO4)3 Æ 24H2O](aq) {(NH4, K)2SO4 Æ Cr2(SO4)3 Æ 24H2O mixed crystals} [63], at T = 298.15 K. The proposed approach has been applied in this study to thermodynamic simulation of[m1(NH4)2SO4 Æ CuSO4 + m2(NH4)2SO4 Æ MgSO4](aq) system at T = 298.15 K. The simulation has been performed as follows: (1) determination on the basis of the (NH4)2SO4(aq), CuSO4(aq), and MgSO4(aq) pure electrolyte (table 1) and mixing h(NH4,Cu), W(NH4,Cu,SO4), h(NH4,Mg), and W(NH4,Mg,SO4) (table 3) parameters, as well as the experimental solubility data on the solubility in the ternary [m1(NH4)2SO4 Æ CuSO4 + m2(NH4)2SO4 Æ MgSO4] (aq) solution, of the component activities in their saturated binary and ternary solution; (2) calculation (equation (16)) on the basis of the experimentally determined composition of the mixed crystals, of the Gibbs free energy of mixing of (NH4)2SO4 Æ (Cu,Mg)SO4 Æ 6H2O crystals, crystallizing from saturated ternary solution. The thermodynamic model for the {m1(NH4)2SO4 Æ CuSO4 + m2(NH4)2SO4 Æ MgSO4}(aq) system was developed on the basis of the experimental composition of saturated solutions and mixed crystalline phase, determined by Hill et al. [79]. There are no data on the mixing {h(Cu,Mg) and W(Cu,Mg,SO4)} parameters for (m1CuSO4 + m2 MgSO4)(aq). According to the experimental solubility

C. Christov / J. Chem. Thermodynamics 37 (2005) 1036–1060

Dmix G
m ðsÞ=ðkJ  mol1 Þ ¼ ð6:1  0:3Þx1 x2 :

ð20Þ

The theoretical basis [48] and the applicability [43,48,51] of a new simplified model for calculation of Dmix G
m ðsÞ have been presented in previous papers by the author. This model allows determination of Gibbs energy of mixing on the basis of experimental data on the solubilities in saturated binary and ternary solutions only, i.e., without any data on the composition of the mixed crystalline phase. When this model is applied, the Dmix G
m ðsÞ values are determined by the change in chemical potential of each of the components during its transition from the so-called ‘‘hypothetic’’ eutonic solution to a saturated ternary solution and

0 -0.2 -0.4 o -1 ∆mixG m/(kJ.mol )

data [108], solid solutions are formed in this ternary mixture. Thermodynamic simulation of the systems with component co-crystallization has shown that neglecting the ternary interactions in the solutions, i.e., assuming h(Cu,Mg) = W(Cu,Mg,SO4) = 0.00, is possible. This approach has been proposed by Pitzer and Kim [3] and has been used for simulation of systems where mixed crystals are formed by anhydrous simple salts [45,46], by crystalline hydrates [45], and by double salts [44,49,63]. In the study of [m1(NH4)2SO4 Æ CuSO4 + m2(NH4)2SO4 Æ MgSO4](aq) system, zero values for the mixing h(Cu,Mg), and W(Cu,Mg,SO4) parameters were accepted. Table 8 shows the calculated component activities ai(l) in the saturated ternary [m1(NH4)2SO4 Æ CuSO4 + m2(NH4)2SO4 Æ MgSO4](aq) solution at T = 298.15 K. The variation in Dmix G
m ðsÞ with the composition of the solid phase is given in last column in table 8 and in figure 5. The mixed (NH4)2SO4 Æ (Cu,Mg)SO4 Æ 6H2O crystals show small positive deviations from the ideal mixed crystals (dashed line on figure 5: f1 = f2 = 1). Obviously, the Dmix G
m ðsÞ values can be described by a dependence which is very close to symmetric and has a minimum at xi  0.5. Using the regular mixing model [43] the Dmix G
m ðsÞ data may be summarized by the following simplified equation:

1055

-0.6 -0.8 -1 -1.2 -1.4 -1.6 -1.8 0

0.2

0.4

0.6

0.8

1

x {(NH4)2SO4.CuSO4.6H2O} FIGURE 5. Plot of the Gibbs free energy of mixing Dmix G
m ðsÞ against the mole fraction x of (NH4)2SO4 Æ CuSO4 Æ 6H2O in (NH4)2SO4 Æ (Cu,Mg)SO4 Æ 6H2O mixed crystals at T = 298.15 K: symbols, calculated values according to equation (16) from experimental data of Hill et al. [79]; - - - - - -, Dmix Gid m (f1 = f2 = 1).

TABLE 9 Pitzer mixing solution parameters at T = 298.15 K calculated by using the Zdanovskii rule (see references [48,158]) System

h

w

Reference

RbCl + CsCl + H2O KCl + RbCl + H2O KBr + RbBr + H2O KI + RbI + H2O NH4Cl + NH4Br + H2O RbCl + RbBr + H2O CsCl + CsBr + H2O MgCl2 + MgBr2 + H2O MgSeO4 + NiSeO4 + H2O MgSeO4 + CoSeO4 + H2O ZnSeO4 + NiSeO4 + H2O ZnSeO4 + CoSeO4 + H2O

0.00025 0.00007 0.00076 0.00100 0.00006 0.00001 0.00010 0.00110 0.022 0.027 0.000 0.000

0.00060 0.00001 0.00005 0.00005 0.00002 0.00001 0.00001 0.00020 0.0028 0.0118 0.0000 0.0008

[43] [48] [48] [48] [48] [48] [48] [43] [51] [51] [51] [51]

TABLE 8 Calculated values of the logarithm of component activities ai(l) in the saturated ternary solution (m1(NH4)2SO4 Æ CuSO4 + m2(NH4)2MgSO4)(aq), and standard molar Gibbs free energy of mixing Dmix G
m ðsÞ of (NH4)2SO4 Æ (Cu,Mg)SO4 Æ 6H2O crystals at T = 298.15 K, where xi is the molar fraction of component i in the mixed crystals Liquid phase

Solid phase

m1/(mol Æ kg )

m2/(mol Æ kg )

ln a1(l)

ln a2(l)

x1a

x2a

Dmix G
m =ðkJ  mol1 Þ

0.000 0.143 0.269 0.361 0.521 0.668 0.783

0.789 0.657 0.536 0.442 0.279 0.128 0.000

0.000 12.97 12.393 12.149 11.868 11.704 11.63

10.20 10.425 10.682 10.927 11.473 12.340 0.000

0.000 0.190 0.368 0.501 0.719 0.891 1.000

1.000 0.810 0.632 0.499 0.281 0.108 0.000

0.000 1.092 1.458 1.549 1.314 0.737 0.000

1 a

a

From Hill et al. [79].

1 a

1056

TABLE 10 Thermodynamic excess ½GE1 ðsÞ (equation (19)) and mixing [A(s)] (equation (18)) parameters for solid solutions at T = 298.15 K System

Mixed crystals

GE1 ðsÞ=ðkJ  mol1 Þ Calculated using equation (16)a

Calculated using equation (21)a 4.264 3.325 3.021 2.517 3.688 2.690 4.340 4.315

KCl + RbCl + H2O

(K,Rb)Cl

3.0 ± 0.5 [164]

4.1 ± 0.5 [45,46]

KBr + RbBr + H2O KI + RbI + H2O NH4Cl + NH4Br + H2O RbCl + RbBr + H2O CsCl + CsBr + H2O MgCl2 + MgBr2 + H2O

(K,Rb)Br (K,Rb)I NH4(Cl,Br) Rb(Cl,Br) Cs(Cl,Br) Mg(Cl,Br)2 Æ 6H2O

2.2 ± 0.6 [165] 1.7 [166,167] 4.2 ± 0.7 [168] 3.1 ± 0.4 [169] 3.5 ± 0.3 [170]

3.0 ± 0.4 [45,46] 2.5 ± 0.5 [45,46] 3.5 ± 0.5 [45,46] 2.8 ± 0.7 [45,46] 4.3 ± 0.7 [45,46] (4.28 ± 0.2) [43]

System

Mixed crystals

A(s)/(kJ Æ mol1)

MgSO4 + NiSO4 + H2O MgSO4+ZnSO4 + H2O ZnSO4 + NiSO4 + H2O MgSeO4 + NiSeO4 + H2O MgSeO4 + CoSeO4 + H2O ZnSeO4 + NiSeO4 + H2O ZnSeO4 + CoSeO4 + H2O K2SO4 Æ NiSO4 + K2SO4 Æ CuSO4 + H2O (NH4)2SO4 Æ CuSO4 + (NH4)2SO4 Æ MgSO4 + H2O 2RbCl Æ NiCl2+2RbCl Æ MnCl2 + H2O (NH4)2SO4 Æ Al2(SO4)3 + K2SO4 Æ Al2(SO4)3 + H2O K2SO4 Æ Cr2(SO4)3 + (NH4)2SO4 Æ Cr2(SO4)3 + H2O (NH4)2SO4 Æ Al2(SO4)3 + (NH4)2SO4 Æ Cr2(SO4)3 + H2O MgSO4 + CoSO4 + H2O

(Mg,Ni)SO4 Æ 7H2O (Mg,Zn)SO4 Æ 7H2O (Zn,Ni)SO4 Æ 7H2O (Mg,Ni)SeO4 Æ 6H2O (Mg,Co)SeO4 Æ 7H2O (Zn,Ni)SeO4 Æ 7H2O (Zn,Co)SeO4 Æ 7H2O K2SO4 Æ (Ni,Cu)SO4 Æ 6H2O (NH4)2SO4 Æ (Cu,Mg)SO4 Æ 6H2O 2RbCl Æ (Ni,Mn)Cl2 Æ 2H2O (NH4,K)2SO4 Æ Al2(SO4)3 Æ 24H2O (K,NH4)2SO4 Æ Cr2(SO4)3 Æ 24H2O (NH4)2SO4 Æ (Al,Cr)2(SO4)3 Æ 24H2O I: Mg(Co)SO4 Æ 7H2O II: Co(Mg)SO4 Æ 7H2O MgSO4 + FeSO4 + H2O I: Mg(Fe)SO4 Æ 7H2O II: Fe(Mg)SO4 Æ 7H2O MnCl2 + FeCl2 + H2O I: Fe(Mn)Cl2 Æ 4H2O II: Mn(Fe)Cl2 Æ 4H2O a The parenthesized values in the data columns are references.

Experimental data

Calculated using equation (7)

(7.1 ± 0.6) (6.2 ± 0.8) (8.3 ± 0.5) (6.6 ± 0.5) (4.5 ± 0.3) (4.2 ± 0.4) (7.3 ± 0.3)

(7.5 ± 0.6) (7.6 ± 0.7) (7.5 ± 0.4) (5.6 ± 0.4) (6.6 ± 0.6) (6.4 ± 0.4) (7.5 ± 0.2) (6.2 ± 0.5) (6.1 ± 0.3) (3.3 ± 0.5) 13.55 [60] 13.6 [60] 12.2 [60] 5.04 [45] 4.37 [45] 6.7 [65] 5.6 [65] 14.4 [50] 11.0 [50]

5.9 [172] 5.8 [172]

[171] [172] [173] [47] [174] [175] [176]

[45] [45] [45] [51] [51] [51] [51] [44] [67] [49]

[48] [43] [48] [48] [48] [48] [48] [43]

Calculated using equation (11)

5.7 6.2 6.5 6.8

[39] [39] [39] [39]

Solubility data source

[164] [130] [165] [166] [168] [169] [170] [177] Solubility data source

[171] [172] [173] [47] [174] [175] [176] [79] [79] [49] [113] [149] [91] [172] [172] [178] [178] [50] [50]

C. Christov / J. Chem. Thermodynamics 37 (2005) 1036–1060

Experimental dataa

C. Christov / J. Chem. Thermodynamics 37 (2005) 1036–1060

not from the saturated binary solution to the saturated ternary solution (equation (16)). The composition of the hypothetic eutonic is determined assuming that no mixed crystals appear in the system under consideration, and the simple substances (components of the solution) or their crystalline hydrates are in fact crystallizing there [48]. If the regular solution model describes the properties of the system sufficiently well, then the maximum Dmix G
m ðsÞ value may be calculated according to the following equation: Dmix G
m ðsÞ ¼ 0:5RT ln fa1 ðlÞ  a2 ðlÞg=fa1 ðlo Þ  a2 ðlo Þg; ð21Þ where a1(l) and a2(l) are the activities of components in the saturated ternary solution in which the mole fraction of each component is equal to the mole fraction in the hypothetic eutonic. The application of equation (21) is possible if the system under consideration obeys the additivity rule [43]. The necessary conditions in this case are: (i) with respect to the liquid phase the system should follow the rule of Zdanovskii [158] and (ii) with respect to the solid phase the rational activity coefficients (f1 and f2) of the components must be equal to each other for all mixed crystal compositions, i.e., f1/f2 = 1. The case of ideal mixed crystals formation (f1 = f2 = 1) belongs here. The ternary ion-interaction parameters for (m1MgCl2 + m2MgBr2](aq) [43], alkali-halide [43,48], and selenate [51] solutions at T = 298.15 K, were evaluated using the rule of Zdanovskii (see table 9). The compositions of the hypothetic eutonics were then calculated using: (i) the calculated parameters and (ii) neglecting the ternary interactions in the solutions, i.e., assuming h = w = 0. Very good agreement was found [48,51], which supports the assumption that h = w = 0 is possible for the ternary systems under consideration. In references [45,50], a method for calculation of Gibbs free energy of phase transition, DGII–I(s), from one structure (II) to another (I) is proposed for systems in which a discontinuous series of mixed crystals is formed. Using this method, we have established that for the discontinuous series of (Mg,Co)SO4 Æ 7H2O mixed crystals the phase transition from a monoclinic (II) (II), Co(Mg)SO4 Æ 7H2O to a rhombic (I) (I), Mg(Co)SO4 Æ 7H2O structure is thermodynamically more favourable {DGII–I(s) =  0.16 kJ Æ mol1} [45]. For the (Mn,Fe)Cl2 Æ 4H2O mixed crystals, the phase transition from a cis-{(II), Mn(Fe)Cl2 Æ 4H2O} to a trans-{(I),Fe(Mn)Cl2 Æ 4H2O} structure is thermodynamically more favourable {DGII–I(s) = 1.10 1 kJ Æ mol } [50]. Table 10 compares the Dmix G
m ðsÞ and Dmix GEm ðsÞ values, presented in the literature [164–176] with those calculated using equations (16) and (21). The experimental solubility data [164–178] used in Dmix G
m ðsÞ and Dmix GEm ðsÞ calculations are given in the last column.

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Thermodynamic excess {GE(s)} and mixing (A) parameters, summarized in table 10, are defined in equations (18) and (19), respectively.

4. Summary and restrictions The equilibrium model presented in this article calculates (solid + liquid) equilibria in the (m1CuCl2 + m2NiCl2)(aq), (m1MgCl2 + m2CrCl3)(aq), (m1(NH4)2SO4 + m2CuSO4)(aq), and (m1(NH4)2SO4 + m2MgSO4)(aq) systems at T = 298.15 K and to high solution concentration within experimental uncertainty. The mixed solution model parameters h(MN) and w(MNX) have been chosen on the basis of the compositions of saturated ternary solutions and data on the pure water solubility of the (NH4)2SO4 Æ CuSO4 Æ 6H2O and (NH4)2SO4 Æ MgSO4 Æ 6H2O double salts. The component activities of the saturated (m1(NH4)2SO4 Æ CuSO4 + m2(NH4)2SO4 Æ MgSO4)(aq) and in the mixed crystalline phase were determined and the change of the molar Gibbs free energy of mixing Dmix G
m ðsÞ of crystals was determined as a function of the solid phase composition. It is established that at T = 298.15 K the mixed (NH4)2SO4 Æ (Cu,Mg)SO4 Æ 6H2O crystals show small positive deviations from the ideal mixed crystals. The results of standard molality-based Pitzer ion interaction model thermodynamic studies at T = 298.15 K on 62 binary, 82 ternary and eight multicomponent (water + salt) systems where solid phases with a constant stoichiometric composition (simple and double salts) crystallize have been summarized. The standard Gibbs free energy of formation Df G
m of the 66 simple and 62 double salts crystallizing from saturated binary and ternary solutions have been calculated. The Df G
m values obtained are in very good agreement with those available in the literature. Values for the integral Gibbs free energy of mixing Dmix G
m ðsÞ and excess Gibbs free energy of mixing DGEm ðsÞ in crystals, calculated by different methods, for the 23 ternary systems with isomorphic or isodimorphic co-crystallization of the salt components have been tabulated. To evaluate the pure electrolyte parameters, we used initial activity data for binary solutions only, i.e., without any complex systems solubility or activity data adjustments. The mixed solution model parameters have been chosen on the basis of the compositions of saturated ternary solutions and data on the pure water solubility of the double salts. The model parameterization is tested by comparison of the predicted and experimental composition of the invariant points in multicomponent solutions (not used in parameterization process).

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JCT 04-210