A Pitzer Parametrization To Predict Solution Properties and Salt

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

pubs.acs.org/jced

A Pitzer Parametrization To Predict Solution Properties and Salt Solubility in the H−Na−K−Ca−Mg−NO3−H2O System at 298.15 K Adeline Lach,*,† Laurent André,†,‡ Sylvain Guignot,† Christomir Christov,§ Pierre Henocq,∥ and Arnault Lassin† †

Water, Environment and Ecotechnologies Division, BRGM, 3 Avenue C. Guillemin, 45060 Orléans Cedex, France Université d’Orléans, CNRS, BRGM, UMR 7327 Institut des Sciences de la Terre d’Orléans, 45071 Orléans, France § Konstantin Preslavsky University, Shumen, 9700, Bulgaria ∥ Research and Development Division, Andra, 1-7 Rue Jean Monnet, Parc de la Croix Blanche, 92298 Chatenay-Malabry Cedex, France ‡

S Supporting Information *

ABSTRACT: A new set of Pitzer interaction parameters is proposed to describe the excess properties of the H−Na−K−Ca−Mg−NO3− H2O system at 298.15 K. From these parameters we reproduce the osmotic coefficient, mean activity coefficients, and density for the binary systems, and also salt solubility in the ternary subsystems. The binary interaction parameters are either selected from the literature or redetermined in this study. In the case of KNO3−H2O and HNO3− H2O binary systems, partial electrolyte dissociation has to be taken into account to represent accurately all the experimental data up to very high concentrations. Then solubility data of salts in the corresponding ternary systems are used to determine the ternary interaction parameters and the solubility products of the double salts. As KNO3 and HNO3 are partially dissociated, we have determined new interaction parameters involving neutral species (for instance ηKNO3/Na+/H+ or μHNO3/KNO3/Na+) to draw the phase diagram of quaternary systems Na−K−H−NO3. To complement the study, we also propose a set of volumetric Pitzer interaction parameters for all the binary systems and for the H−K−NO3 ternary system, so solution density can be computed at 298.15 K.

1. INTRODUCTION Saline solutions, from brackish water to brines, are commonly encountered or generated in large-scale industrial operations such as drinking water production, geothermal energy recovery,1 CO2 storage in deep geological aquifers,2 and the extraction of valuable mineral compounds and elements such as Na2SO4, KOH, Si, Li, B, and Br.3 To implement or improve the processes underlying such operations we need accurate prediction of the brines’ reactivity in the various corresponding environments. Geochemical modeling can be a useful tool to evaluate their physical and chemical properties in line with reactivity, such as water activity, aqueous species activity, or solubility products. However, a prerequisite for such modeling is the establishment of an often complex thermodynamic framework that accounts for the high ionic strengths of the brines, the interaction between aqueous species, and the resulting substantial departures from ideal solution behavior. Over the years, various approaches4−7 have been developed for understanding the thermodynamics of saline solutions. Among them, and relying on the original work of Debye and Hückel,8 whose equations are valid for low molalities, the semiempirical Pitzer model4,9−13 has an extended field of applications. It can model the excess properties of electrolyte solutions up to very high salinities14,15 since it considers binary © XXXX American Chemical Society

and ternary interactions between charged and noncharged aqueous species in addition to the effects of ionic strength. Originally developed for a fully dissociated single electrolyte,4 the model knew several improvements and can now describe complex (i.e., multielectrolyte) aqueous systems, including partially dissociated electrolytes, and for concentrations up to several tens of moles per kilogram of water. The related equations are described in many publications,10,11,14−16 and they are sufficiently robust to describe the properties of aqueous electrolyte mixtures up to these concentrations. Our study aims to provide a set of Pitzer interaction parameters to determine the thermodynamic properties of the H−Na− K−Ca−Mg−NO3−H2O chemical system at 298.15 K and up to the solubility of salts. Though parts of this chemical system have been investigated,17,18 the full system has never been studied despite its interest in many applications. Nitrates are expected to play a significant role in the context of the underground repository of nuclear waste. More precisely, long-lived intermediatelevel radioactive waste that is planned to be stored in deep clay formations are composed of dried sludge from effluent Received: November 2, 2017 Accepted: February 12, 2018

A

DOI: 10.1021/acs.jced.7b00953 J. Chem. Eng. Data XXXX, XXX, XXX−XXX

Journal of Chemical & Engineering Data

Article

others, and also considered HNO3-containing systems, providing an internally consistent set of Pitzer parameters for the H−Na−K−Ca−Mg−NO3−H2O chemical system at 298.15 K, and valid up to the solubility of salts.

treatments that contain substantial quantities of nitrate among other elements. They are enclosed in specific containers that are placed in underground cavities dug in argillite host rock with very low permeability. Storage safety analyses show that, despite the protection of the concrete or stainless steel-made external container layers, the formation water of the host rock is likely to migrate and reach the waste during the disposal period. This would result in the potential dissolution of large amounts of nitrate and of other species, resulting in highly saline, corrosive, and oxidative media with high reactivity toward the containment materials and their surroundings, including the host rock. This reactivity must therefore be characterized, and this work is a first step toward this objective. Some subsystems have been already studied. For instance, Kalinkin17 determined Pitzer parameters for the H−Na−K− NO3−H2O system and introduced β(2) ca specific interaction parameters for the binary systems to reproduce correctly solubility data (in the three ternary systems and the quaternary system). Originally, the β(2) ca parameter was introduced to describe 2:2 type electrolyte solutions at low concentrations,10 but later it was used to improve the description of other types of electrolytes characterized by very high solubility19,20 while maintaining the hypothesis of full dissociation of the electrolytes. Steiger 21,18 studied the Na−K−NO 3 and Na−Mg−NO3 systems. He also introduced β(2) ca parameters and, in addition, optimized the α1 and α2 parameters (associated with the β(1) ca and β(2) ca interaction parameters, respectively) to reproduce experimental data up to solubility but also in the supersaturation domain. In the original Pitzer model, α1 = 2 and α2 = 0 for any electrolyte (except 2:2 electrolytes for which α1 = 1.4 and α2 = 12). Marion22 proposed Pitzer interaction parameters for the H−Na−K−Mg−Ca−NO3 system for low temperatures (< −73.15 to 298.15 K). Using another approach that distinguishes short-range and long-range interactions, but still considering full electrolyte dissociation (except for CaCl2 and MgCl2), Gruszkiewicz et al.23 described a large part of the Na−K−Mg−Ca−Cl−NO3 system as a function of temperature. But some ternary systems including Ca (e.g., Ca−K−NO3) were still an issue since salt solubility is not described on the whole range of compositions. In this work we addressed this latter ternary system in addition to

2. THEORETICAL BACKGROUND Excess Gibbs Free Energy. The Pitzer model expresses the excess Gibbs free energy of an aqueous solution (Gex). By deriving the excess Gibbs free energy with respect to the mass of water (ww) or to the species mole number, we obtain the osmotic coefficient of the solution (ϕ) and the species activity coefficient (γi), respectively. For a system containing c cations, a anions, and n neutral species, the excess Gibbs free energy is computed following eq 1.24 ⎧ 4IAϕ Gex /(wwRT ) = ⎨ ln(1 + bI1/2) − ⎪ b ⎩ ⎪

+ 2 ∑ ∑ mc ma(Bca + (∑ mc zc)Cca) c

+

Bca Cca Φcc′, Φaa′ ψcc′a, ψaa′c λni ζnca ηncc′, ηnaa′ μnni, μnn′i

∑∑ c

+ +

c

mc mc (2ΦVcc ′

c′

′

+

∑ maψccV a) ′

a

∑ ∑ mama ′(2ΦVaa ′ + ∑ mcψaaV c) + ∑ mn2λnn a

a′

∑

′

c

mn3μnnn

n

n

+ 2 ∑ ∑ mnmn λnn ′ ′ n n′

+ 3 ∑ ∑ mn2mn μnnn + 6 ∑ ∑ ∑ mnmn mn″μnn n ″ ′ ′ ′ ′ n n′ n n′ n″ + 2 ∑ ∑ mnmc λnc + 3 ∑ ∑ mn2mc μnnc n

c

n

c

+ 2 ∑ ∑ mnmaλna + 3 ∑ ∑ mn2maμnna n

+ +

a

n

a

∑ ∑ ∑ mnmcmaζnca + ∑ ∑ ∑ mnmcmc′ηncc n

c

a

∑ ∑ ∑ mnmcma ′ηnaa n

a

a′

n

′

c

c′

′

+ 6 ∑ ∑ ∑ mc mnmn μcnn ′ ′ c n n′

⎫ + 6 ∑ ∑ ∑ mamnmn μann ⎬ ′ ′⎪ ⎭ a n n′ ⎪

Table 1. Summary and Description of the Pitzer Specific Interaction Parameters Potentially Involved in the Modelling of a Complex Aqueous System Pitzer parameter

a

(1)

where R is the ideal gas constant, T is the absolute temperature, Aϕ is the Debye−Hückel parameter, I is the ionic strength, b is a universal parameter set to 1.2 by Pitzer,4 mi is the molality of species i, zc is the charge of the cation c. All these parameters are in SI units. All other parameters are specific interaction parameters and are summarized in Table 1. Partial Dissociation. Taking the partial dissociation into account implies correcting the calculated osmotic coefficient in order to compare it to the literature data.16,25,26 Indeed, the values of experimental osmotic coefficient reported in the literature were obtained considering total dissociation. Therefore, the calculated osmotic coefficient ϕ must be converted as if total dissociation was accounted for. The invariant parameter is the water activity (this measurement does not depend on solution speciation):

comment (1) (2) c/a (cation/anion) binary interactions; contain β(0) ca , βca , βca , α1, and α2 parameters c/a binary interactions; contain the Cϕca parameter cation−cation or anion−anion ternary interactions; contain the θcc′ and θaa′ parameters, respectively ternary interactions between one ion and two oppositely charged ions binary interactions between one neutral species, n, and itself (i = n) another neutral (i = n′) or charged species (i = c or a) ternary interactions between one neutral species, one cation and one anion ternary interactions between one neutral species and two likecharge ions ternary interactions between one neutral species, n, and itself (i = n), another neutral species (i = n′) or one charged species (i = c or a). Also possible: interactions between two different neutral species, and a third neutral species or a cation or an anion

ln a w ′ = ln a w

(2)

where aw′ is the water activity considering partial dissociation and aw is the water activity considering total dissociation. If we B



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where V0i is the standard partial volume of species i at infinite dilution computed using the Helgeson−Kirkham−Flowers equations.32 By integrating Gex computed according to the Pitzer model in eq 7, Vϕ can be computed using eq 8. This equation means that the partial electrolyte dissociation, if any, is explicitly taken into account for calculating the solution density.

consider the osmotic coefficient, the following correction is obtained: ϕ′ = ϕ

∑i mi ∑i νimi

(3)

where ϕ′ is the converted osmotic coefficient, νi is the number of ions formed by the complete dissociation of one molecule of solute, and mi is the molality of solute in mol·kg−1. A correction is also necessary for the mean activity coefficient (γ±): ln γ±′ = f ln γ±

Vϕ =

∏ aiν /K

+ + +

i

+

∑ maψccV a) ′

a

∑ ∑ mama ′(2ΦVaa ′ + ∑ mcψaaV c) + ∑ mn2λnnV a′

∑

′

c

V mn3μnnn

mnmn λnnV

+ 2∑∑ n

n

′

n′

c

n

+ 2∑∑ n

+ +

mnmaλnaV

n

′

c

n

a

∑∑∑ n

a

c

V + 3 ∑ ∑ mn2maμnna

a

∑∑∑ n

a′

V mnmc ma ηnca ′

′

V mnmc ma ηnaa ′

′

a

+

∑ ∑ ∑ mnmcmc′ηnccV n

c

c′

′

V + 6 ∑ ∑ ∑ mc mnmn μcnn ′ ′ c n n′

⎫ V ⎬ + 6 ∑ ∑ ∑ mamnmn μann ⎪ ′ ′⎭ a n n′ ⎪

(8) −1

−1

Here R′ is the ideal gas constant in cm ·bar·mol ·K ; AV is the Debye−Hückel coefficient for volume (linked to the pressure derivative of Debye−Hückel coefficient) in cm3·kg1/2·mol−3/2 and YV represents any volumetric interaction parameter (i.e., BVca, λVnn, ζVnca, ...), analogous to the specific interaction parameters summarized in Table 1. More details can be found in Lach et al.30 Parameterization. For some chemical systems, the interactions parameters found in the literature are not consistent or are not valid up to high concentrations, so we had to evaluate a new set of interaction parameters. The parametrization methodology is based on the coupling between the geochemical calculation code PhreeSCALE30 and the optimization software PEST.33 This is an iterative method able to fit calculated values to experimental data. Its principle is shown in Figure 1. The interaction parameters of binary systems are optimized from osmotic coefficient data. The ternary interaction parameters are optimized based on solubility data. The best set of interaction parameters is the one which leads to a saturation ratio of 1. The closer SR is to unity, the better is the parametrization. The solubility constants of salts are determined from the calculation of the ionic activity product at the experimental equilibrium molality of the binary system. For the solubility constant of double salts, the value is optimized during the parametrization process of ternary systems. 3

(6)

+ [∂Gex /∂P]T , m /∑ ni

c′

′

V + 2 ∑ ∑ mnmc λncV + 3 ∑ ∑ mn2mc μnnc

where ρw is the density of pure water, and Mi represents the molar mass of species i. Vϕ is the apparent molar volume computed from pressure derivative of the excess Gibbs energy following eq 7: ∑i mi

′

V + 6 ∑ ∑ ∑ mnmn mn″μnnV n ″ + 3 ∑ ∑ mn2mn μnnn ′ ′ ′ ′ n n′ n n′ n″

31

Vϕ =

c

mc mc (2ΦVcc

n

1000 + ∑i miMi

∑i miV i0

∑∑ a

(5)

Vϕ ∑i mi + 106(ρw )−1

a

c

where K is the solubility constant of the mineral, νi is the stoichiometric coefficient of the species i in the dissolution/precipitation reaction, and ai its activity. At equilibrium, the ionic activity product (∏i aiνi ) is equal to the solubility constant, and SR = 1. This parameter is of great importance in determining the equilibrium constant of minerals when the solubility is known. Computation of Density Using the Pitzer Model. The Pitzer model can also be used to compute other properties such as heat capacity,27 excess enthalpy,27 or density.28 The first two of those are linked to the temperature derivatives of the excess Gibbs energy, and density is related to the pressure derivative of the excess Gibbs energy. Two distinct sets of interaction parameters28−30 are necessary to compute, on the one hand, osmotic coefficient, activity coefficient, excess enthalpy, and heat capacity and, on the other hand, density. The density of aqueous solution ρ is computed following eq 6: ρ=

∑i mi c

i

i

⎪

+ 2 ∑ ∑ mc ma(BcaV + (∑ mc zc)CcaV )

(4)

where f is the dissociation fraction, which is calculated as the ratio of the mole number of dissociated product to the initial mole number. Liquid−Solid Equilibria. In studies of liquid−solid equilibria, the saturation ratio (SR), defined by eq 5, is a widely used parameter that measures the equilibrium deviation for a given mineral: SR =

⎛ R′T ⎞⎧ IA ln(1 + bI1/2) ⎟⎟⎨ V + ⎜⎜ ⎪ b ⎝ ∑i mi ⎠⎩ R′T

∑i miV i0

(7)

Figure 1. Principle of optimization method. C



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Table 2. Binary Interaction Parameters at 298.15 K values at 298.15 K β(0) Na+/NO−3

3.798586 × 10−03

β(1) Na+/NO−3

ref 22

σ

values at 298.15 K

0.00373

β(0) Mg+2/NO−3

3.286010 × 10−01

1.835089 × 10−01

β(1) Mg+2/NO−3

1.915868 × 1000

C(ϕ) Na+/NO3− β(0) K+/NO−3 β(1) K+/NO−3 C(ϕ) K+/NO−3 (0) βH+/NO−3 β(1) H+/NO−3 C(ϕ) H+/NO−3

0

C(ϕ) Mg+2/NO3− β(0) Ca+2/NO−3 β(1) Ca+2/NO−3 C(2) Ca+2/NO−3

−6.353680 × 10−03

α2

4.705000 × 10−01

C(ϕ) Ca+2/NO−3

−2.401200 × 10−03

λ

4.125504 × 10−02

− HNO3/NO3

−01

−1.610270 × 10

0.00914

8.675356 × 10

2.217643 × 10−02 −01

1.065751 × 10

this study

0.00642

−01

4.583727 × 10

1.151500 × 10−01

σ 0.01182

this study

0.01113

1.505170 × 1000 1.403660 × 10−01

2.187520 × 10−03

λHNO3/H

4.125504 × 10−02

λHNO3/HNO3

6.727369 × 10−02

μHNO3/HNO3/HNO3

−9.051000 × 10−04

+

this study

−01

ref 34

3. RESULTS 3.1. Solubility. 3.1.1. Binary Systems. The binary interaction parameters are either provided by the literature or optimized from osmotic coefficient data. Their values and the standard deviation obtained on the osmotic coefficient experimental data are reported in Table 2. NaNO3−H2O. The interaction parameters of NaNO3 electrolyte are taken from Marion.22 Indeed, in a previous study,35 we investigated the solubility of darapskite in the Na−NO3−SO4 ternary system, and a review of the existing set of parameters for the binary NaNO3−H2O system confirms that the Marion’s set is valid up to nitratine (NaNO3) solubility (Figure 2). To be Figure 3. Comparison between experimental and modeling data of the Mg(NO3)2-H2O binary system at 298.15 K for the mean activity coefficient (left) and osmotic coefficient (right). ○, ref 37; △, ref 40; −, This work; ---, nitromagnesite (Mg(NO3)2·4H2O) solubility.

domain. Consequently, this system is revisited to take into account data available in the supersaturation domain (over 8 mol·kg−1). This is necessary because in some ternary systems (see the K−Ca−NO3 ternary system) the molalities of Ca(NO3)2 can be higher than the solubility limit of the binary system. The interaction parameters were optimized on the osmotic coefficient data measured by El Guendouzi and Marouani,37 Platford,43 Figure 2. Comparison between experimental and modeling data of the NaNO3−H2O binary system at 298.15 K for the mean activity coefficient (left) and osmotic coefficient (right). △, ref 36; ○, ref 37; □, ref 38; −, this work; ---, nitratine (NaNO3) solubility.

consistent with our previous study, we continue here to use this parameter set. Mg(NO3)2−H2O. For the Mg(NO3)2-H2O system, the parameters proposed by Rard et al.34 were selected. Indeed, with these parameters a standard deviation of 0.01182 is obtained against 0.18819 with the parameters of Kim and Fredericks,39 0.16564 with the parameters of Marion22 and 0.01281 with the parameters of Steiger.18 In Figure 3, we plotted the activity coefficient and the osmotic coefficient of the binary system. This set of parameters is valid up to nitromagnesite (Mg(NO3)2.6H2O) solubility and above. Ca(NO3)2−H2O. The interaction parameters of Ca(NO3)2 available in the literature37,41,42 are not valid in the supersaturation

Figure 4. Comparison between experimental and modeling data of the Ca(NO3)2·H2O binary system at 298.15 K for the mean activity coefficient (left) and osmotic coefficient (right). ○, ref 37; △, ref 43; □, ref 44; ◊, ref 45; −, this work; ---, nitrocalcite (Ca(NO3)2·4H2O) solubility; -·-, Ca(NO3)2·3H2O. D



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Many osmotic coefficient data are reported in the literature (El Guendouzi and Marouani;37 Hamer and Wu36 and Robinson and Stokes40) and some authors (El Guendouzi and Marouani37) proposed interaction parameters able to describe the osmotic coefficient data with good accuracy up to niter (KNO3) solubility (i.e., 3.8 mol·kg−1). However, as described later (see section 3.1.2), in some ternary systems, KNO3 molality can reach values far beyond the limit of solubility in the binary system. The consequence is having to extend the validity of the interaction parameters over the solubility limits in the binary system. When extension is impossible with classical interaction parameters (β(2) ca , α1, α2), the solution is to introduce partial electrolyte dissociation (here KNO3) and consider the ion pair (here KNO30) in the calculations. We determine the binary interaction parameters (including the KNO30 ion pair) by considering osmotic coefficient data simultaneously with the ternary interaction parameters relative to the KNO3−Ca(NO3)2−H2O ternary system, also using solubility data (see section 3.1.2). For the binary system, the osmotic coefficient and the mean activity coefficient (corrected using eq 3 and 4, respectively) are plotted in Figure 5. The binary interaction parameters are reported in Table 2 and the equilibrium constant for the reaction formation of the ion pair KNO30 is given in Table 3. In Figure 6, we also plotted the aqueous speciation of KNO3, illustrating its dissociation rate as a function of total KNO3 concentration in water. At the solubility limit in the binary system (i.e., 3.8 mol·kg−1), about 23% of KNO3 is nondissociated, meaning that the solution contains 2.93 mol·kg−1 of K+ (and also of NO3− owing to the electroneutrality constraint in the aqueous solution) and 0.87 mol·kg−1 of KNO30.

Figure 5. Comparison between experimental and modeling data of the KNO3−H2O binary system at 298.15 K for the mean activity coefficient (left) and osmotic coefficient (right). ○, ref 37; △, ref 36; □, ref 40; −, this work; ---, niter (KNO3) solubility.

Robinson,44 and Stokes.45 In Figure 4, we plotted the osmotic coefficient and the mean activity coefficient for the binary system. The solubilities of the stable nitrocalcite salt (Ca(NO3)2· 4H2O) and of the less hydrated metastable phase Ca(NO3)2· 3H2O are also shown on Figure 4. The interaction parameters used are reported in Table 2. For this binary system, we reproduced experimental data without considering partial dissocia(2) tion, but with optimized βCa and α2 parameters. The α1 +2 /NO− 3

parameter is set to 2.0, as in the original model. KNO3−H2O. The case of the binary KNO3−H2O system is much more complex than that of the previous systems.

Table 3. Equilibrium Constants at 298.15 K for Mineral Dissolution Reactions and for Ion Pairing Reaction, Deliquescence Relative Humidity (DRH) at Saturation and Standard Gibbs Energy of Formation ΔG°f (kJ/mol)

DRH, % compound

msat (exp)

reaction

KNO30 HNO30 Nitratine (NaNO3)

K+ + NO3− = KNO30 H+ + NO3− = HNO30 NaNO3 = Na+ + NO3−

Niter (KNO3)

KNO3 = K+ + NO3−

Nitrocalcite (Ca(NO3)2·4H2O) Ca(NO3)2·4H2O = Ca2+ + 2NO3− + 4H2O

Ca(NO3)2·3H2O Ca(NO3)2·KNO3·3H2O nitromagnesite (Mg(NO3)2· 6H2O) KNO3·2HNO3 K+ Na+ Ca2+ Mg2+ H+ NO3− H2O a

Ca(NO3)2·3H2O = Ca2+ + 2NO3− + 3H2O Ca(NO3)2·KNO3·3H2O = Ca2+ + K+ + 3NO3− + 3H2O Mg(NO3)2·6H2O = Mg2+ + 2NO3− + 6H2O KNO3·2HNO3 = K+ + H+ + 3NO3−

10.878 10.860 10.904 3.714 3.768 3.75 8.232 8.409 8.211 8.538 15

4.856 4.883

(aSw × 100)

log K

ref

calc

51

73.76

0 −1.261 1.0864

a a 22

−366.6573

−367

51 52 51 53 51 53 51

92.61

−0.0676

a

−393.8009

−394.86

49.27

1.969

a

−1711.938

−1713.15

2.32 1.848

a a

−1472.794 −1868.903

−1471.7

3.05

a

−2082.617

−2080.6

5.98

a 54 54 54 54 54 54 54

−581.093 −282.51 −261.953 −552.806 −455.375 0 −110.905 −237.14

ref

53 51

53.50

ref data50

This study. E



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demonstrate that the two constraints (osmotic coefficient and partial dissociation) can be successfully described up to very high concentrations. All these parameters are used to study the ternary systems in order to maintain internal consistency in the final set of interaction parameters and equilibrium constants. 3.1.2. Ternary systems. The ternary interaction parameters (θcc′, ψcc′a, ζnca, λnc, ... see eq 1 and Table 1) were determined only from the solubility data (except for the KNO3−Ca(NO3)2− H2O system, as described in section 3.1.1). To ensure the consistency of the thermodynamic database beyond the nitrate system, we took the interaction parameter θcc′ from the literature since it has already been defined for other ternary systems (such as c−c′−Cl and c−c′−SO4). Only the θCa+2/Mg+2 parameter was optimized because, in a recent study,49 the interaction parameters of the CaCl2−H2O and MgCl2−H2O binary systems were revised, and consequently, the values of the θCa+2/Mg+2 parameter found in the literature no longer made sense. Table 3 lists the equilibrium constants for mineral dissolution reactions and the deliquescence relative humidity (in %). The deliquescence relative humidity represents the water activity in the brine generated when water vapor starts to condensate at the salt surface. Below this value, no condensation occurs because the brine is not stable; above, the salt dissolves. This property is important for the salt conservation and used for the humidity extractors. Table 3 also includes the standard Gibbs energy of formation (ΔGf°) of the various chemical compounds involved in the study. For the salts, the values determined in this work are compared to the values of Wagman et al.50 A maximum deviation of 2 kJ is obtained for the nitromagnesite (Mg(NO3)2· 4H2O) between the literature and computed values. We compared the standard deviation obtained with our interaction parameters with those in the literature on the solubility experimental data. For each investigated system, the standard deviation is better than in previous studies, confirming the consistency of this new parametrization. Table 4 reports the selected and optimized values of ternary interaction parameters. NaNO3−KNO3−H2O. The experimental solubility data in this ternary system were compiled by Silcock.51 They reported three experimental studies. From these experimental data, we optimized three ternary interaction parameters: ψK+/Na+/NO−,

Figure 6. Calculated change in K+ and KNO30 concentration as a function of total KNO3 molality at 298.15 K. −, KNO30 molality; -·-, K+ molality; ---, niter (KNO3) solubility.

HNO3−H2O. For the HNO3−H2O binary system, experimental measurements of the HNO3 dissociation rate in water exist.46−48 These measurements prove the existence of a neutral species, which is therefore integrated in the model. Consequently, we used these data, in addition to the osmotic and activity coefficient data,36 to constrain the determination of a relevant set of binary interaction parameters (Table 2) and the equilibrium constant of the formation reaction of the HNO30 ion pair. In Figure 7, the osmotic coefficient and the mean activity

Figure 7. Comparison between experimental and modeling data of the HNO3−H2O binary system at 298.15 K for the mean activity coefficient (left) and the osmotic coefficient (right) up to 30 mol·kg−1: △, ref 36; −, this work.

3

ζKNO3 /Na+/NO−3 and λKNO3/Na+ (Table 4). In Table 3, we reported the values of the solubility constant of nitratine (NaNO3) and niter (KNO3). The results are plotted in Figure 9. NaNO3−HNO3−H2O. Nitratine (NaNO3) solubility is plotted as a function of HNO3 molality in Figure 10. Experimental data (symbols) are taken from Friese and Ebel.62 Two parameters (ψNa+/H+ /NO− and λHNO3/Na+) are necessary to reproduce 3 the experimental solubility data (Table 4). NaNO3−Ca(NO3)2−H2O. Silcock51 compiled experimental solubility data for the NaNO3−Ca(NO3)2−H2O ternary system (two experimental studies, so two invariant points). Only one interaction parameter (ψNa+/Ca+2 /NO−) is optimized so that the 3 experimental data are represented correctly (Figure 11). Our model represents the experimental data over the whole range of molality. NaNO3−Mg(NO3)2−H2O. The experimental data in this system were reported by Silcock51 and Benrath63 and are consistent with each other. Only one interaction parameter (ψNa+/Mg +2 /NO−) is necessary to reproduce the experimental 3 data (Figure 12).

coefficient for the binary system are plotted, while Figure 8 shows the HNO 3 dissociation fraction. These figures

Figure 8. Dissociation fraction of HNO3 at 298.15 K: ref 47; ○, ref 48; −, this work.

△,

ref 46; ◇, F


Journal of Chemical & Engineering Data Table 4. Interaction Parameters and Standard Deviation in Ternary Systems at 298.15 K parameters

values at 298.15 K

ref

σa

θK+/Na+ ψK+/Na+/NO−3

−3.203 × 10−03 2.5746 × 10−05

Na−K−NO3 System 55 this study 0.01758b

ζKNO3/Na+/NO−3

8.2068 × 10−03

0.11032c

λKNO3/Na

+

θH+/Na+ ψNa+/H+/NO−3 λHNO3/Na

+

−01

1.207 × 10

0.01677

−02

−2.0 × 10

θNa+/Ca+2 ψNa+/Ca+2/NO−3

5.0 × 10−02 −5.0 × 10−03

θNa+/Mg+2 ψNa+/Mg+2/NO−3

7.0 × 10−02 −1.8 × 10−02

0.01636

c

Na−Ca−NO3 System 57 0.04135b this study 0.30384c

θH+/K+ ψK+/H+/NO−3

Na−Mg−NO3 System 58 0.02754b this study 0.06055c 0.04222d K−H−NO3 System 1.538 × 10−02 56 0.01619b −2.824 × 10−02 this study 2.65735c

λHNO3/K+

−1.013 × 10−01

λKNO3/H+

7.742 × 10−02

λHNO3/KNO3

1.578 × 10−01

θK+/Ca+2 ψK+/Ca+2/NO−3

K−Ca−NO3 System 1.156 × 10−01 55 0.04701b −3.424 × 10−02 this study 0.89542c

λKNO3/Ca+2

−1.033 × 10−01

ηKNO3/K+/Ca2+

−3.540 × 10−02

σa

n

parameters

55

μKNO3/KNO3/Ca2+

2.291 × 10−03

θK+/Mg+2 ψK+/Mg+2/NO−3

K−Mg−NO3 System 0 58 0.09521b −02 −4.549 × 10 this study 0.22880c

λKNO3/Mg+2

2.312 × 10−01

d

Na−H−NO3 System 3.4538 × 10−02 56 −8.0 × 10−03 this study 0.01333b

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8

values at 298.15 K

ηKNO3/K /Mg

−2.651 × 10−01

6 19

45

16

66 51

Ca−H−NO3 System 59 0.10910b this study 0.45619c

θH+/Ca+2 ψCa+2/H+/NO−3

9.686 × 10−02 9.636 × 10−03

λHNO3/Ca+2

1.006 × 10−01

μHNO3/HNO3/Ca2+

−5.040 × 10−03

ηHNO3/H+/Ca2+

−3.387 × 10−03

θCa+2/Mg+2 ψCa+2/Mg+2/NO−3

Ca−Mg−NO3 System −5.776 × 10−02 this study 0.07437b −1.234 × 10−03 0.14078c

θH+/Mg+2

1.000 × 10−01

ψCa+2/Mg+2/NO−3

−2.760 × 10−03

ζHNO3/Mg+2/NO−3

1.117 × 10−03

0.15773b

λHNO3/Mg+2

1.878 × 10−01

5.37719c

H−Mg−NO3 System 60

μHNO3/HNO3/Mg2+

1.810 × 10−03

ηHNO3/H /Mg

−5.440 × 10−02

+

2+

11

0.25721d

5.319 × 10−01

2+

n

K−Ca−NO3 System

ζKNO3/Mg+2/NO−3 +

ref

17

14 12 29

this study

σ is the standard deviation ( (Y exp − Y mod)2 /n ) achieved after parameter optimization for each ternary chemical system; n is the number of experimental data points used in the optimization process. bThis study. cReference 22. dReference 21, 61.

a

Figure 9. Calculated (lines) and experimental (symbols) solubility in the NaNO3−KNO3−H2O ternary system at 298.15 K. Filled symbols represent the invariant point. ○ and ●, Nikolaev (1928) in ref 51; □ and ■, Karnaukhov (1956) in ref 51; ◇ and ◆, Cornec and Krombach (1929) in ref51; −, nitratine solubility; ---, niter (KNO3) solubility.

Figure 10. Calculated (lines) and experimental (symbols) solubility in the NaNO3−HNO3−H2O ternary system at 298.15 K. ○, ref 62; −, nitratine (NaNO3) solubility

this value, we plotted in Figure 13 the calculated hypothetical invariant equilibrium point between this double salt and niter (KNO3) (dotted line). It is obtained for KNO3 and HNO3 concentrations of 4.54 and 17.36 mol·kg−1, respectively. By decreasing the KNO3 content and increasing the HNO3 content in the solution it is then possible to draw a hypothetical solubility curve for the double salt, thus defining its stability domain. Kalinkin et al.17 plotted a partial phase diagram for the NaNO3−KNO3−HNO3−H2O system, which suggests that the invariant point of KNO3·2HNO3 and niter (KNO3) is obtained for KNO3 and HNO3 concentrations around of 4.2 and

KNO3−HNO3−H2O. Niter (KNO3) solubility is plotted as a function of HNO3 molality in Figure 13. Experimental data (symbols) are taken from Flatt and Bocherens64 and Linke.65 Four parameters ( ψK+/H+ /NO−, λ HNO 3 /K + , λ KNO 3 /H + and 3

λHNO3/KNO3) are necessary to reproduce the experimental solubility data of niter (KNO3; Figure 13). In the Na−K− H−NO3 quaternary system (see section 3.1.3), the equilibrium constant of the KNO3·2HNO3 double salt is determined. Using G



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room temperature is suggested. Consequently, there is not enough information to conclude definitively on this system and we plotted only a partial phase diagram. According to our model, the experimental solubility of niter (KNO3) measured for HNO3 concentrations above 17.41 mol·kg−1 would correspond to metastable states. KNO3−Ca(NO3)2−H2O. Silcock51 compiled experimental data for this ternary system. They reported two mineral phases: nitrocalcite (Ca(NO3)2·4H2O) and niter (KNO3). Flatt and Bocherens52 reported the existence of the double salt Ca(NO3)2·KNO3·H2O. The values reported in the two references for the nitrocalcite (Ca(NO3)2·4H2O) and niter (KNO3) stability domains are similar so we could use the whole set of experimental solubility data. As mentioned earlier (see section 3.1.1), this ternary system was parametrized simultaneously with the binary KNO3−H2O system, including the binary interaction parameters and the formation constant of the KNO30 ion pair. For the ternary system, five parameters are necessary to represent the experimental data: ψK+/Ca+2 /NO−, λKNO3/Ca+2, 3 ηKNO /K+/Ca 2+ , μKNO /KNO /Ca 2+ and the solubility constant of

Figure 11. Calculated (lines) and experimental (symbols) solubility in the NaNO3−Ca(NO3)2−H2O ternary system at 298.15 K. Filled symbols represent the invariant point. ○ and ●, Kremann and Rodemund (1914) in ref 51; ◇ and ◆, Pinaevskaya et al. in ref 51; −, nitratine (NaNO3) solubility; ---, nitrocalcite (Ca(NO3)2·4H2O) solubility.

3

3

3

the double salt (Figure 14). This diagram clearly shows that the

Figure 12. Calculated (lines) and experimental (symbols) solubility of NaNO3−Mg(NO3)2−H2O ternary system at 298.15 K. Filled symbols represent the invariant point. ○ and ●, Sieverts and Muller (1930) in ref 51 ; ◇ and ◆, ref 63; −, nitratine (NaNO3) solubility; ---, nitromagnesite (Mg(NO3)2·4H2O) solubility.

Figure 14. Calculated (lines) and experimental (symbols) solubility in the KNO3−Ca(NO3)2−H2O ternary system at 298.15 K. Filled symbols represent the invariant point. ○ and ●, ref 52; △ and ▲, Hamid and Das (1930) in ref 51; −, niter (KNO3) solubility; ---, Ca(NO3)2·KNO3·3H2O solubility; -·-, nitrocalcite (Ca(NO3)2·4H2O) solubility.

hypothesis of partial dissociation improves the model significantly since the solubility of the niter (KNO3) is correctly represented in this case. To our knowledge this is the first time this system has been successfully described in the literature. KNO3−Mg(NO3)2−H2O. The experimental data for this system are reported by Silcock.51 All data are consistent. Four interaction parameters (ψK+/Mg +2 /NO−, λKNO3/Mg+2, ζKNO3 /Mg 2+/NO−3 3

and ηKNO3/K+/Mg2+) are necessary to reproduce the experimental data (Figure 15). In addition to the solubility data, experimental isopiestic data can be used to constrain the parametrization. Todorovic and Ninković68 used the isopiestic method to measure the osmotic coefficient of the KNO3−Mg(NO3)2−H2O ternary system, KNO3 being used as the reference salt. The measurements were made for different ionic strength fractions of KNO3 computed as yKNO3= mKNO3/(mKNO3 + 3mMg(NO3)2). The results are plotted in Figure 16 as a function of ionic strength. They match the experimental data satisfactorily. Ca(NO3)2−HNO3−H2O. The Ca(NO3)2-HNO3−H2O ternary system was experimentally studied by Flatt and Fritz.69 Other

Figure 13. Calculated (lines) and experimental (symbols) solubility in the KNO3−HNO3−H2O ternary system at 298.15 K. ◇, ref 64; ○, Nikolaev (1928) in ref 65; ∗, invariant point computed in this work; − and ---, niter (KNO3) solubility; -·-, KNO3·2HNO3 solubility.

17.2 mol·kg−1, respectively, when NaNO3 is equal to 0. Linke,66 on the basis of the work of Nikolaev et al.,67 plotted the phase diagram for the ternary system HNO3−KNO3−H2O. Their diagram suggests that at 298.15 K, it is actually the double salt KNO3·HNO3 that forms and that the invariant point is higher in HNO3 concentration (around 170 mol·kg−1). But the acquisition of these data is unclear. Moreover, Linke66 reported other studies where the presence of the double salt KNO3·HNO3 at H



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shows some discrepancies with the experimental Ca(NO3)2· 3H2O solubility data. Some improvements could be made by considering the existence of other salts such as Ca(NO3)2· 2H2O and Ca(NO3)2 quoted by different authors. However, their respective domain of stability is close or above 30 mol·kg−1 of HNO3, a domain of concentration not explored in this paper due to the lack of data. Ca(NO3)2−Mg(NO3)2−H2O. Two sets of experimental data are reported in the literature. First, Matveeva and Kudryashova70 report the formation of Ca(NO3)2·3H2O in the ternary system, at the expense of nitrocalcite (Ca(NO3)2·4H2O) and nitromagnesite (Mg(NO3)2·4H2O). However, Yin et al.53 did not observe this solid phase at 298.15 K, and they proposed the existence of a single invariant point corresponding to the equilibrium between nitrocalcite (Ca(NO3)2·4H2O) and nitromagnesite (Mg(NO3)2·4H2O). Second, Platford43 reported experimental iso-water activity data for Ca(NO3)2−Mg(NO3)2 mixtures of various ionic strengths. From the model viewpoint, the solubility data of Matveeva and Kudryashova70 are well represented. If the data of Yin et al.53 are used to optimize the ternary interaction parameters, it becomes possible to represent either nitromagnesite (Mg(NO3)2·4H2O) solubility (taking ψCa+2 /Mg +2 /NO− =

Figure 15. Calculated (lines) and experimental (symbols) solubility in the KNO3−Mg(NO3)2−H2O ternary system at 298.15 K. Filled symbol represents the invariant point. ◇ and ◆, Benrath and Sichelschmidt (1931) in ref 51; lines, this work; −, niter (KNO3) solubility; ---, nitromagnesite (Mg(NO3)2·4H2O) solubility.

3

−0.009) or nitrocalcite (Ca(NO3)2·4H2O) solubility (taking

Figure 16. Calculated (lines) and experimental (symbols) osmotic coefficient of KNO3−Mg(NO3)2-H2O ternary system at 298.15 K as a function of ionic strength: inverted ⬠, y = 0; ◊, y = 0.2015; ▽, y = 0.3967; △, y = 0.5941; ○, y = 0.7934; □, y = 1; y represented the ionic strength fraction of KNO3 (see text) ref 68; −, this work.

experimental solubility data were reported by Silcock.51 Four parameters (ψCa+2 /H+ /NO−, λHNO3/Ca+2, μHNO3/HNO3/Ca+2 and

Figure 18. Solubility and iso-water activity compositions in the Ca(NO3)2−Mg(NO3)2−H2O ternary system at 298.15 K. ○ and ●, ref 53; ∗, invariant point;70 ◇, aw = 0.977;43 ◆, aw = 0.893;43 △, aw = 0.822;43 ▲, aw = 0.693;43 inverted ⬠, aw = 0.641;43 -··-, nitromagnesite (Mg(NO3)2·4H2O) solubility; ---, Ca(NO3)2·3H2O solubility; -·-, nitrocalcite (Ca(NO3)2·4H2O) solubility.

3

ηHNO3/H+/Ca+2) are necessary to represent as well as possible the experimental data (Figure 17). Our model represents correctly

Figure 17. Calculated (lines) and experimental (symbols) solubility in the Ca(NO3)2−HNO3−H2O ternary system at 298.15 K. ○ and ◇, Basset and Taylor (1912) ref 51; ● and ◆, ref 69; ∗, invariant point69 −, nitrocalcite (Ca(NO3)2·4H2O) solubility; ---, Ca(NO3)2·3H2O solubility.

Figure 19. Calculated (lines) and experimental (symbols) osmotic coefficient of Ca(NO3)2−Mg(NO3)2−H2O ternary system at 298.15 K as a function of ionic strength fraction of Ca(NO3)2. The ionic strength varies between □, 1.38 and 1.53; ○, 4.96 and 6.49; △, 7.22 and 10.02; ▽, 10.71 and 15.79; ◇, 12.05 and 18.09 (ref 43); −, this work.

the nitrocalcite (Ca(NO3)2·4H2O) solubility data, up to a total content in HNO3 of about 17 mol·kg−1. Beyond that, the model I



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ψCa+2 /Mg +2 /NO− = −0.003), but there are no parameters to 3

represent all the data sets. Therefore, we optimized the parameters on the basis of the two former experimental data sets, 70 using only one ternary interaction parameter, ψCa+2 /Mg +2 /NO−. Results are plotted in Figure 18. In Figure 19, 3

we plotted the osmotic coefficient as a function of ionic strength fraction of Ca(NO3)2: yCa(NO3)2 = 3mCa(NO3)2/ (3mCa(NO3)2+3mMg(NO3)2). Mg(NO3)2−HNO3−H2O. Nitromagnesite (Mg(NO3)2·4H2O) solubility is plotted as a function of HNO3 molality in Figure 20. Experimental data (symbols) are taken from Ewing and Klinger71 and evidence surprising behavior above 25 mol·kg−1 of HNO3, since two regions of salt-solution equilibrium can be identified. One extends at moderate concentrations of Mg(NO3)2, that is, below 6 M, and the other extends at high Mg(NO3)2 concentrations, above 9 M. Because of this particular behavior, five parameters (ψMg +2 /H+ /NO−, ζHNO3 /Mg +2 /NO−3 , λHNO3/Mg+2,

Figure 20. Calculated (lines) and experimental (symbols) solubility of Mg(NO3)2−HNO3−H2O ternary system at 298.15 K. ○, ref 71; −, nitromagnesite (Mg(NO3)2·4H2O) solubility.

Table 5. Parameters for Complex Systems at 298.15 K parameters

values

ref

μKNO3/HNO3/Na+

−0.0034819

this study

ηKNO3/Na+/H+

−0.0215731

this study

3

μHNO3/HNO3/Mg2+ and ηHNO3/H+/Mg2+) are necessary to reproduce the experimental solubility data (Figure 20), at least for the region of moderate Mg(NO3)2 concentrations. Our model is able to represent experimental data up to 35 mol·kg−1 while previous models22 are limited to 6 mol·kg−1 of HNO3. 3.1.3. Na−K−H-NO3 System. The experimental data of the Na−K−H−NO3 system are reported by Silckock.51 We optimize two new parameters (μKNO3/HNO3/Na+ and ηKNO3/Na+/H+) and the solubility constant for the double salt KNO3·2HNO3 on the basis of the experimental solubility data provided by Silcock.51 Values obtained are reported in Table 5. In Figure 21, we plotted a “partial” phase diagram for the quaternary system Na−K−H−NO3 at 298.15 K. This diagram is called “partial” because no data on the invariant points for the ternary systems KNO3−HNO3−H2O and NaNO3−HNO3−H2O have been found in the literature. 3.2. Density. 3.2.1. Binary Systems. A set of volumetric interaction parameters was determined in addition to the set of Pitzer interaction parameters used to describe osmotic and activity coefficients. To build this new set of parameters we relied upon experimental density data either taken from the literature or acquired in this study (see Appendix 1). The interaction parameter values are reported in Table 6. Table 7 lists the sources of all experimental data. All measured data are consistent with each other and consequently, for the convenience of representation, we plotted only some experimental data on the graph. The results obtained are plotted in Figure 22 to Figure 26.

Figure 21. Solubility of salts (in mass fraction but without considering the solvent) in the quaternary system NaNO3−KNO3−HNO3−H2O at 298.15 K. ◇, Niter (KNO3) + nitratine (NaNO3) invariant point (Nikolaev (1928) in ref 51) □, nitratine (NaNO3) + KNO3·2HNO3 (Nikolaev (1928) in ref 51), ∗, invariant point computed in this work; −, model; this work.

Table 6. Volumetric Binary Interaction Parameters at 298.15 K values at 298.15 K −6

β(0)V Na+/NO3− β(1)V Na+/NO−3 C(ϕ)V Na+/NO−3 β(0)V K+/NO−3 β(1)V K+/NO−3 C(ϕ)V K+/NO−3 VKNO30

4.4792 × 10

β(0)V Mg+2/NO3−

1.1313 × 10−5

β(1)V Mg+2/NO−3 C(ϕ)V Mg+2/NO−3

ref

values at 298.15 K

this study

9.211 × 10−5 0 −2.961 × 10−5

this study

−4

7.4680 × 10 0 13.411

this study

0

−6

β(0)V Ca+2/NO3− β(1)V Ca+2/NO−3 β(ϕ)V Ca+2/NO−3 β(0)V H+/NO−3 β(1)V H+/NO−3 C(ϕ)V H+/NO−3 λVHNO3/HNO3

4.8082 × 10

λVKNO3/H+

4.4792 × 10−6

λVKNO3/HNO3

9.211 × 10−5

ref this study

3.189 × 10−4 0 −9.761 × 10−6

this study

−5

2.9297 × 10 0

−3.465 × 10−6 this study

0 J



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Table 7. Experimental Density of Nitrate Systems System

ref

NaNO3−H2O KNO3−H2O HNO3−H2O Ca(NO3)2−H2O Mg(NO3)2−H2O

72−81 72−74, 77−79, 82, 83 74, 77, 84, 85 77, 86, 87 77, 81, 88, 89

Figure 25. Calculated (lines) and experimental (symbol) density for the Ca(NO3)2−H2O binary system at 298.15 K: ○, ref 86; ◇, ref 87; −, this work.

Figure 22. Calculated (lines) and experimental (symbol) density for the NaNO3−H2O binary system at 298.15 K: △, ref 72; ○, ref 75; ◇, ref 79; −, this work.

Figure 26. Calculated (lines) and experimental (symbol) density for the Mg(NO3)2−H2O binary system at 298.15 K: ○, ref 88; ◇, ref 81; △, ref 89; −, this work.

ternary interaction parameters were determined (λVKNO3/H+ and λVKNO3/HNO3). Values obtained are reported in Table 7. In Figure 27, we plotted experimental and calculated density as a function of HNO3 molality. Figure 23. Calculated (lines) and experimental (symbol) density for the KNO3−H2O binary system at 298.15 K: ○, ref 83; ◇, ref 79; △, this work; −, this work.

4. CONCLUSIONS We have proposed a new Pitzer parametrization to study the complex H−Na−K−Ca−Mg−NO3−H2O system at 298.15 K. All ternary systems are correctly represented up to the salt solubility and up to 30 mol HNO3·kg−1. For two electrolytes (HNO3 and KNO3), we had to take into account their partial dissociation to accurately represent the data, especially in ternary systems in which electrolyte concentrations are higher than the solubility limits established in the binary system. For HNO3, the presence of HNO3° is due to the high concentrations above 30 mol HNO3·kg−1; moreover, some experimental measurements proved the existence of this neutral species. For KNO3, the hypothesis of a neutral species was introduced to correctly represent the Ca−K−NO3 ternary system. Having to pair ion species means that we must determine specific interaction parameters (ηn/c′/c, ηn′/c′/c and μn/n′/c) in order to model quaternary systems. We also determined volumetric interaction parameters to calculate the density of binary and ternary nitrate systems. The next step is to study this nitrate system in basic conditions and to extend the temperature domain for representing radioactive waste disposal conditions associated to concrete vaults and exothermic HLW (high-level waste), respectively.

Figure 24. Calculated (lines) and experimental (symbol) density for the HNO3−H2O binary system at 298.15 K: ○, ref 85; ◇, ref 74; −, this work.

3.2.2. Ternary System. Some density measurements have been performed on the ternary HNO3−KNO3−H2O system (Appendix 1). To represent the data correctly, two volumetric K



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J.C. Robinet, B. Madé, and P. Henocq). Our thanks go to Karen M. Tkaczyk who polished the English in the manuscript.

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(1) Gupta, H. K.; Roy, S.; Gupta, H. K. Geothermal Energy : An Alternative Resource for the 21st Century; Elsevier, 2007. (2) Metz, B.; Davidson, O.; Coninck, H. De.; Loos, M.; Meyer, L. IPCC Special Report on Carbon Dioxide Capture and Storage; C. U. Press: New York, 2005. (3) Bourcier, W.; Lin, M.; Nix, G. Recovery of Minerals and Metals from Geothermal Fluids; Lawrence Livermore National Laboratory: CA, 2005. (4) Pitzer, K. S. Thermodynamics of Electrolytes. I. Theoretical Basis and General Equations. J. Phys. Chem. 1973, 77, 268−277. (5) Chen, C.; Britt, H. I.; Boston, J. F.; Evans, L. B. Local Composition Model for Excess Gibbs Energy of Electrolyte Systems. Part I: Single Solvent, Single Completely Dissociated Electrolyte Systems. AIChE J. 1982, 28, 588−596. (6) Li, J.; Polka, H.-M.; Gmehling, J. A gE Model for Single and Mixed Solvent Electrolyte Systems: 1. Model and Results for Strong Electrolytes. Fluid Phase Equilib. 1994, 94, 89−114. (7) Zhao, E.; Yu, M.; Sauvé, R. E.; Khoshkbarchi, M. K. Extension of the Wilson Model to Electrolyte Solutions. Fluid Phase Equilib. 2000, 173, 161−175. (8) Debye, P.; Hückel, E. Zur Theorie Der Elektrolyte. I. Gefrierpunktserniedrigung Und Verwandte Erscheinungen. Phys. Zeitschrift 1923, 24, 185−206. (9) Pitzer, K. S.; Mayorga, G. Thermodynamics of Electrolytes. II. Activity and Osmotic Coefficients for Strong Electrolytes with One or Both Ions Univalent. J. Phys. Chem. 1973, 77, 2300−2308. (10) Pitzer, K. S.; Mayorga, G. Thermodynamics of Electrolytes. III. Activity and Osmotic Coefficients for 2−2 Electrolytes. J. Solution Chem. 1974, 3, 539−546. (11) Pitzer, K. S.; Kim, J. J. Thermodynamics of Electrolytes. IV. Activity and Osmotic Coefficients for Mixed Electrolytes. J. Am. Chem. Soc. 1974, 96, 5701−5707. (12) Pitzer, K. S. Activity Coefficients in Electrolyte Solutions, 2nd ed.; CRC Press: Boca Raton, 1991. (13) Zemaitis, J. F.; Clark, D. M.; Rafal, M.; Scrivner, N. C. Handbook of Aqueous Electrolyte Thermodynamics; John Wiley & Sons: New York, 1986. (14) Lassin, A.; Christov, C.; André, L.; Azaroual, M. A Thermodynamic Model of Aqueous Electrolyte Solution Behavior and Solid-Liquid Equilibrium in the Li-H-Na-K-Cl-OH-LiCl0(aq)H2O System to Very High Concentrations (40 Molal) and from 0 to 250°C. Am. J. Sci. 2015, 315, 204−256. (15) Lach, A.; André, L.; Lassin, A.; Azaroual, M.; Serin, J.-P.; Cézac, P. A New Pitzer Parameterization for the Binary NaOH-H2O and Ternary NaOH-NaCl-H2O and NaOH-LiOH-H2O Systems up to NaOH Solid Salt Saturation, from 273.15 to 523.15 K and at Saturated Vapour Pressure. J. Solution Chem. 2015, 44, 1424−1451. (16) Pitzer, K. S.; Silvester, L. F. Thermodynamics of Electrolytes. VI. Weak Electrolytes Including H3PO4. J. Solution Chem. 1976, 5, 269− 278. (17) Kalinkin, A. M. Calculation of Phase Equilibria in the NaNO3KNO3-HNO3-H2O System at 25°C over the Range from 0 to 20 M HNO3. Russ. J. Appl. Chem. 1996, 69, 664−668. (18) Steiger, M. The Geochemistry of Nitrate Deposits: I. Thermodynamics of Mg(NO3)2−H2O and Solubilities in the Na+− Mg2+−NO3−−SO42‑−H2O System. Chem. Geol. 2016, 436, 84−97. (19) Beyer, R.; Steiger, M. Vapour Pressure Measurements and Thermodynamic Properties of Aqueous Solutions of Sodium Acetate. J. Chem. Thermodyn. 2002, 34, 1057−1071. (20) Christov, C. Thermodynamics of Formation of Double Salts and Mixed Crystals from Aqueous Solutions. J. Chem. Thermodyn. 2005, 37, 1036−1060. (21) Steiger, M.; Kiekbusch, J.; Nicolai, A. An Improved Model Incorporating Pitzer’s Equations for Calculation of Thermodynamic

Figure 27. Calculated (lines) and experimental (symbol) density for the HNO3−KNO3−H2O ternary system at 298.15 K: ◆, KNO3 = 0 mol·kg−1; ◇, KNO3 = 0.5 mol·kg−1; ●, KNO3= 1 mol·kg−1; ○, KNO3= 1.5 mol·kg−1; ▼, KNO3= 2 mol·kg−1; ▽, KNO3= 2.5 mol·kg−1; ×, KNO3 = 3 mol·kg−1; inverted ⬠, KNO3= 3.5 mol·kg−1; ▲, KNO3= 4 mol·kg−1; △, KNO3= 5 mol·kg−1; ■, KNO3= 6 mol·kg−1; □, KNO3= 7 mol·kg−1; ∗, KNO3= 8 mol·kg−1; inverted ⬟, KNO3 = 9 mol·kg−1, −, model; this work, ---, model, in the supersaturation domain.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jced.7b00953. Experimental density for the HNO3−KNO3−H2O ternary system at 298.15 K (PDF)

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REFERENCES

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Adeline Lach: 0000-0001-7651-8242 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This research was carried out as part of the BRGM-Andra scientific partnership (CTEC laboratory group, coordinated by L



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