# Understanding the Solid Solution–Aqueous Solution Equilibria in the KCl + RbCl + H_{2}O System from Experiments, Atomistic Simulation and Thermodynamic Modeling

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## Abstract

The solid solution–aqueous solution (SSAS) equilibria in the KCl + RbCl + H_{2}O system were redetermined at 298.15 K. The experimental data for (K,Rb)Cl were consistent with the formation of a continuous solid solution without miscibility gaps. The Schreinemakers’ wet residues method and an XRD quantitative analysis technique based on the Vegard approach were applied to determine the chemical composition of the solid solution phase of (K,Rb)Cl. The compositions of (K,Rb)Cl derived from the Vegard approach are in accordance with those from the wet residues method. The thermodynamic properties of mixing of the (K,Rb)Cl solid solution were theoretically predicted using atomistic simulations. From these simulations, a regular solution behavior is recognized that is consistent with the knowledge of the thermodynamic properties of mixing of (K,Rb)Cl obtained from SSAS equilibrium studies, but the predicted regular solution model parameter *A* _{0} is significantly larger than that regressed from the SSAS equilibrium data. Finally, a thermodynamic model was developed for representing the SSAS equilibria and element partitioning in the KCl + RbCl + H_{2}O system as a function of temperature that can be used for predicting the SSAS equilibria in the studied system over the temperature range 273.15–373.15 K.

## Keywords

KCl + RbCl + H_{2}O Solid solution Vegard’s rule Atomistic simulation Thermodynamic modeling

## 1 Introduction

Solid solution–aqueous solution (SSAS) equilibria have been the subject of numerous industrial, geochemical and environmental investigations [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]. The term *solid solution* was introduced by van’t Hoff, and to date solid solution is sometimes referred to as mixed crystal [17]. When two solutes crystallize simultaneously from an aqueous phase with similar structures, a solid solution is likely to form [7]. Information on the SSAS equilibria is of particular importance for the purification of chemical products [2, 15], material synthesis [18], trace element partitioning [14, 19, 20, 21], and studies of the contamination of soils and aquifers [7] and global element cycles.

The study of SSAS equilibria and thermodynamics has attracted much attention from physical chemists since the late 19th century as a means of obtaining high-purity chemical regents [22, 23, 24, 25]. Several decades later, the special importance of SSAS equilibria in geochemistry and environmental science was also emphasized, and several seminal reviews on the phase diagram representation, thermodynamic modeling and kinetic characteristics of SSAS equilibria have been published [2, 4, 5, 7, 14, 26]. In the late 19th century, the physical chemist W. Nernst gave a general mathematical description, which is widely known as Nernst’s distribution law, of the distribution of a solute between two immiscible solvents [23]. Doerner and Hoskins [24] first applied the SSAS equilibria when studying the “co-precipitation” of BaSO_{4} and RaSO_{4} from an aqueous solution and the “replacement” of BaSO_{4} in a mixed radium-barium chloride solution. To represent the partitioning of Ra and Ba between the solid and aqueous phases, they proposed the partition law and introduced the distribution constant *K*, which has been widely applied in trace element geochemistry.

At the end of the 19th century, Roozeboom [25] described five possible types of solid solution in systems consisting of two salts and water using a special graphical representation known as the Roozeboom diagram corresponding to the commonly used Gibbs triangular diagram [1, 17]. Later, Ricci [17, 27] suggested a possible sixth type of solid solution formation of two components in the ternary aqueous system Na_{2}SO_{4}−NaBrO_{3}−H_{2}O. This sixth type of solid solution was again reported by Campbell et al. in the Li_{2}SO_{4}−LiBrO_{3}−H_{2}O system [28], but its existence is still under debate. In addition to the fundamental Gibbs phase diagram and Roozeboom diagram, Lippmann [26] proposed a quasi-phase diagram to simultaneously represent the liquid and solid compositions for a binary solid solution in equilibrium with an aqueous solution by defining the total solubility product (\( \varSigma \varPi \)) and activity mole fraction (*x* _{act}) of the substituting ion in the aqueous solution [2, 5, 29]. The total solubility product, \( \varSigma \varPi \), of a binary solid solution A_{ x }B_{1–x }C is defined as the sum ({A^{+}}+{B^{+}}){C^{−}} of the partial solubility products of each end member (AC and BC) as expressed on an activity scale. In a typical Lippmann diagram, two superimposed abscissas—the activity mole fraction of the substituting ion in the aqueous solution *x* _{act} = {B}/({A} + {B}) and the mole fraction of the end member in the solid solution *x* _{B}—are plotted as x-axes, and \( \varSigma \varPi \) is plotted as the y-axis. Similar to the Lippmann diagram, Königsberger [13] proposed the *φ* \( \varSigma \) *m* diagram representation of ternary SSAS equilibria. In this representation, *x* _{act} and *x* _{B}, with the same meanings as in the Lippmann diagram, are also plotted as superimposed x-axes, while *φ* \( \varSigma \) *m* is plotted as the y-axis, where *φ* is the osmotic coefficient of the aqueous solution saturated with a solid solution, and \( \varSigma \) *m* is the total molality of the saturated solution.

To completely represent the phase equilibria in the SSAS system, it is necessary to simultaneously know the compositions of the aqueous and solid phases. By filtering and performing an analysis, the composition of the solid solution can be obtained, but the result is usually not sufficiently accurate, as some mother liquor is inevitably left in the filtered solid phase. To determine the compositions of the dry solid phases, the wet residues method was introduced by F. A. H. Schreinemakers [17, 30]. Schreinemakers’ wet residues method is quite effective for ternary systems, but for more complex systems, it is seldom applied. Even in a ternary system, this method meets difficulty in determining the phase compositions when two solid solutions are simultaneously saturated with an aqueous solution, as only the average composition of the two solid solutions can then be given. Therefore, for systems containing more than one solid solution phase, only the liquid phase composition is usually reported.

Theoretical treatments, especially thermodynamic modeling, for the SSAS equilibria have made significant progress since the 1980s [2, 3, 4, 5, 7, 13, 29, 31, 32, 33, 34, 35]. Using the “stoichiometric saturation” concept, Thorstenson and Plummer simulated the non-equilibrium dissolution thermodynamics of a binary solid solution compound Ca_{ x }Mg_{1–x }CO_{3} in aqueous solution [31]. Later, Glynn and Reardon [2] reviewed the “stoichiometric saturation” model and provided a new derivation of the relation between the solid-phase component activity and stoichiometric saturation constant *K* _{ss}. They also derived the relation between *K* _{ss} and the excess Gibbs energy *G* ^{E} for a A_{ x }B_{1−x }C-type solid solution and used Guggenheim’s sub-regular function to represent *G* ^{E} for a given binary solid solution. Moreover, Glynn and Reardon also suggested that only relatively insoluble solids achieve stoichiometric saturation states when they interact with an aqueous solution, while highly soluble solid solutions, e.g., alkali halides, can reach the thermodynamic equilibrium state after just three to four weeks at 25 °C. Gamsjaeger and Königsberger also made important contributions through SSAS equilibrium studies involving experiments, thermodynamic theory and modeling [13, 36, 37]. They derived the criteria of the stoichiometric saturation equilibrium condition and stable phase equilibrium condition for binary SSAS systems using a unified thermodynamic framework [13]. Based on these theoretical advances, several computer programs, e.g., SOLISOL [33], PASSIPHIC [38], PHREEQC [39], EQ3/6 [40], ChemSage [41] and GEMS-PSI [35, 42, 43], have been developed for simulating SSAS equilibria based on the law of mass action (LMA) method [39] and the Gibbs Energy Minimization (GEM) approach [13, 35].

For any specific SSAS system, knowledge of *K* _{ss} or *G* ^{E} is necessary for modeling the SSAS reaction and equilibria. *K* _{ss} is usually determined by fitting the experimental solubility data of a single stoichiometric solid solution, while *G* ^{E} is determined from the solubility data of solid solutions with varying compositions. Hence, the composition of the solid solution is of vital importance for the thermodynamic modeling of the SSAS equilibria. Glynn developed a code, named MBSSAS, for regressing the parameters in the *G* ^{E} equation from SSAS equilibrium data [3]. Alternative approaches were introduced by Christov et al. to determine the regular or sub-regular parameters from SSAS equilibrium data [20, 21]. To date, *G* ^{E} equation parameters for some binary solid solutions in ternary salt–water systems have been determined at 298.15 K [3, 21].

To reveal the structure–property relations and predict the thermodynamic properties of solid solutions, atomistic simulation approaches have been introduced in recent years [8, 10, 44, 45, 46]. These purely theoretical approaches were first used in alloy systems for predicting the thermodynamic quantities of alloy phases and providing necessary data for phase diagram simulations [44, 45, 47, 48, 49]. This type of methodology is usually called ab initio-aided CALPHAD thermodynamic modeling [50]. Vinograd et al. [8, 10, 42, 46] expanded the atomistic simulation approaches to mineral and salt solid solutions. Although the thermodynamic quantities predicted by atomistic simulations have been used for phase diagram prediction, these systems were mainly non-aqueous, e.g., the systems Mg_{3}Al_{2}(SiO_{4})_{3}–Ca_{3}Al_{2}(SiO_{4})_{3}, NaCl–KCl, CaCO_{3}–MgCO_{3} and CaCO_{3}–MgCO_{3}–MnCO_{3}. In the case of the SSAS equilibria, only the phase diagrams of the systems CaCO_{3}–SrCO_{3}–H_{2}O and BaSO_{4}–RaSO_{4}–H_{2}O were predicted using ab initio solid solution thermodynamics, but for these systems solid–aqueous equilibrium experimental data have not been reported over the full composition range, which poses a problem for verifying the reliability and accuracy of those predictions thoroughly. Eclectically, the reliability of the model predictions can only be validated by comparison with the available thermodynamic and phase equilibrium data in the corresponding non-aqueous systems, and/or the trace element partition coefficients obtained from laboratory or field experiments.

The ternary KCl + RbCl + H_{2}O system is a typical SSAS system and one of the most important preliminary systems for understanding the solid–aqueous equilibria in the multi-component system Na, K, Rb, Mg//Cl−H_{2}O, which is essential to revealing the trace element Rb partitioning during potash deposit formation and KCl production [21, 51, 52]. Although the phase equilibria of the KCl + RbCl + H_{2}O system have been studied experimentally at several temperatures and simulated at 298.15 K, the complete SSAS equilibria and Rb and K partitioning between the aqueous and solid phases as functions of temperature are still poorly understood. In this work, we redetermined the SSAS equilibria in the KCl + RbCl + H_{2}O system at 298.15 K. The commonly used Schreinemakers’ wet residues method and the inverse Vegard’s rule were applied to determine the solid solution compositions, and the consistency of the two approaches is discussed. We then studied the thermodynamics of the solid solution (K,Rb)Cl with atomistic simulations. Finally, a comprehensive thermodynamic model for representing the SSAS equilibria and element partitioning in the KCl + RbCl + H_{2}O system as a function of temperature was developed.

## 2 Experimental Section

### 2.1 Chemicals Used

^{−4}S·m

^{−1}) obtained from a water purification system (UPT-II-20T, Chengdu Ultrapure Technology Co., Ltd.) was used in all experiments.

Sources, purities and CAS numbers of chemicals

Chemical name | Source | Final mass fraction purity | Purification method | CAS no. |
---|---|---|---|---|

Rubidium chloride | Shanghai Aladdin Reagent Company | >0.9995 | None | 7791—11-9 |

Potassium chloride | China National Pharmaceutical Industry | >0.9999 | Recrystallization | 7447—40-7 |

### 2.2 Experimental Process and Sample Analysis Methods

Six samples were prepared by adding different ratios of KCl, RbCl and H_{2}O into Teflon FEP bottles (approximately 60 mL) and capping. Enough salts were added to ensure that solids were always present during the equilibration period. The bottles were further sealed with Parafilm and placed in an atmospheric thermostatic bath (298 ± 0.5 K), and a magnetic stirrer was used to promote equilibration. The equilibration time lasted one month.

One month later, the magnetic stirring was stopped, and the bottles were kept static for approximately 3 h to separate the supernatant, which was then taken out for analysis using a syringe with a fritted glass filter (pore size 3–4 μm). One part of the supernatant liquid was injected into a 15 mL quartz weighing bottle with cover, evaporated to dryness at approximately 333 K for 10 h and heated at 413 K to constant weight to determine the water content of the sample [53, 54]. Another part of the supernatant liquid was used for analyzing the Cl^{−} content by the gravimetric method of silver chloride precipitation [55]. In each analysis, three parallel samples were taken, and the maximum relative standard uncertainty was found to be less than 0.001 for both the H_{2}O and Cl^{−} contents. From these determinations, the contents of KCl and RbCl in the aqueous solutions can be calculated, and the relative standard uncertainties of the KCl and RbCl contents can be controlled within 0.005.

The chemical compositions of the wet solids (directly taken out using a crystal spoon without further processing) were also determined using the same methods as for the supernatant liquids. Another piece of wet solid was dried with filter papers and characterized using X-ray powder diffraction (PANalytical X’Pert PRO diffractometer with Cu Kα radiation and *λ* = 0.15406 nm) at room temperature. X’Pert HighScore and Jade software were used for the phase identification, cell refinement and lattice parameter calculations.

The scanning electron microscopic (SEM) and energy-dispersive X-ray spectroscopy (EDS) studies were used to determine the chemical composition of the solids and performed on a JSM-5610LV/INCA scanning electron microscope. The sample used for SEM–EDS analysis was the same as that used for XRD analysis. After being dried at room temperature, the powder sample was inlaid into a type of Cu-based conductive polymer. Then, the inlaid particles were polished with sand paper from coarse to fine, until a smoothed cross section appeared. The surface of the sample was not coated. The excitation voltage used was 10 kV. Back-scattered electron image was collected and for a selected particle the EDS element analysis were performed and the chemical information of about ten points in a straight line were determined. The average composition of these points was taken as the composition of the particle.

### 2.3 Determination of Solid Solution Composition with Vegard’s Rule

Vegard’s rule [56, 57] is an empirical formula for describing the relationship between the lattice parameters of a solid solution and its composition and has been proven valid for many different types of solid solutions, e.g., salts, alloys, oxides and silicates.

*a*

_{ n }terms are the lattice parameters of the solid solution and its end members, and

*x*is the mole fraction of end member A.

*V*

_{ n }terms are the unit cell volumes of the solid solution and its end members, and

*x*is the mole fraction of end member A.

Although the samples taken from the solid–aqueous solution equilibrium experiment are not absolutely dry, the diffraction pattern of the dry solid can be captured from XRD characterizations. From the collected powder XRD patterns, the lattice parameters of the dry solids contained in the wet samples can be calculated using the cell refinement technique implemented in the Jade 6.5 software. The lattice parameters *a* _{A} and *a* _{B} of end members A and B can be determined from the XRD patterns of the solids precipitated from binary aqueous solutions using the same process as for the solid solutions.

The work of Merbach and Gonella [58] indicates that the lattice parameter *a* of the (K,Rb)Cl solid solution, which has a face-centered cubic lattice with *a* = *b* = *c* and *α* = *β* = *γ* = 90°, follows Vegard’s rule well. Therefore, we can derive the chemical composition of the (K,Rb)Cl solid solution contained in the wet residue from the XRD-determined lattice parameters based on Vegard’s rule. Later, the chemical composition determined from Vegard’s rule will be compared with that from Schreinemakers’ wet residues method for each (K,Rb)Cl solid solution to verify the validity of the inverse Vegard’s approach for the determination of the solid solution compositions.

## 3 Atomistic Simulations of Mixing Properties of Solid Solution

### 3.1 Density Functional Theory (DFT) Calculations

Here, we consider an isostructural solid solution with the end members A*R* and B*R*, where atoms A and B can be substituted for each other, and *R* denotes atoms that are not directly involved in the mixing. In the case of the solid solution (K,Rb)Cl, the end members are KCl and RbCl, respectively. Both KCl and RbCl possess a face-centered cubic (FCC) lattice with a space group *Fm*3*m* (IT No. 255) at room temperature. A supercell consisting of 2 × 2 × 2 FCC unit cells with 32 K/Rb and 32 Cl atoms, which is similar to that used in the study of Vinograd and Winkler [8] on the NaCl–KCl system, was used in the DFT calculations. A CASTEP distribution [59] is applied to perform the DFT calculations, and the ultrasoft pseudopotentials integrated in CASTEP were used with the general gradient approximation (GGA). The electron–electron exchange and correlation interactions were treated in the framework of the GGA according to the Perdew–Burke–Ernzerhof (PBE) functional. The plane wave expansion cutoff energy was set to 580 eV, and the *k*-point sampling was according to a 2 × 2 × 2 Monkhorst–Pack grid.

*R*or A

*R*) end member in the simulated supercell. Thus, there will be only one specific configuration generated for each single-defect structure when the supercell is subjected to periodic boundary conditions. In the 2 × 2 × 2 supercell of the (K,Rb)Cl solid solution, the chemical formulas of the single-defect structures are K

_{31}Rb

_{1}Cl

_{32}and K

_{1}Rb

_{31}Cl

_{32}, and the corresponding configurations are shown in Fig. 1. SDM only considers the interaction between the atoms of the host and the solute and is exact only in the dilute (

*x*

_{B}→ 0 or

*x*

_{A}→ 0) and high-temperature (

*T*→ ∞) limits.

For the prediction of the mixing thermodynamics of a concentrated solid solution, defect–defect interactions, which may result in chemical ordering, should be considered. When the assumption of infinite dilution is lifted, the defect–defect interactions could be significant. A double-defect structure assumes that two defect atoms (AA or BB) are dispersed in the matrix of a pure (B*R* or A*R*) end member in the simulated supercell. Defect–defect interactions will also be induced in these structures, in addition to the interactions between the atoms of the host and the solute that are induced in single-defect structures. In the 2 × 2 × 2 supercell of the (K,Rb)Cl solid solution, the chemical formulas of the double-defect structures are K_{30}Rb_{2}Cl_{32} and K_{2}Rb_{30}Cl_{32}. Five different configurations with defect-distance in the range of 4.5–11 Å will be generated in the supercells with double defects. Therefore, the energies in the total of 14 configurations without defects and with single-defect and double-defect structures will be predicted from DFT calculations in the 2 × 2 × 2 supercell of (K,Rb)Cl.

*E*

_{m,mix}of the defected configurations can be calculated according to Eq. 3,

*n*is 1 or 31 for single-defect structures and 2 or 30 for double-defect structures.

*E*(structure) is the DFT predicted energy of a specific configuration.

*H*

_{m,mix}of a configuration is calculated according to

*V*is the excess volume of mixing. Although ∆

*V*can be exactly calculated from the volume change of mixing relative to the pure-substance structures, the pressure

*p*was set to zero all the time in the present DFT calculations. Hence,

*H*

_{m,mix}=

*E*

_{m,mix}all the time.

*G*

_{m}

^{E}of a configuration is defined as

*T*is the absolute temperature in Kelvin and

*S*

_{m}

^{E}is the molar excess entropy. For every configuration, all defects are seen as random in the DFT calculations, and there is no explicit excess entropy, that is,

*S*

_{m}

^{E}= 0 and then

*G*

_{m}

^{E}=

*H*

_{m,mix}.

*A*

_{0}and

*A*

_{1}can be determined by fitting these equations to the DFT generated

*G*

_{m}

^{E}of all configurations. In general, Eq. 6 is used unless the

*x*− G

_{m}

^{E}curve is strongly asymmetric.

The parameters *A* _{0} and *A* _{1} are possibly dependent on the temperature and pressure. However, following Vinograd et al. [10], here we will apply the parameters *A* _{0} and *A* _{1} regressed from the DFT mixing thermodynamics of the solid solution (K,Rb)Cl to modeling the SSAS equilibria of the ternary KCl + RbCl + H_{2}O system in Sect. 4.

### 3.2 Monte Carlo Simulations

*J*

_{ r }, is defined as the enthalpy effect of the exchange reaction AA + BB = 2AB at a distance

*r*within the supercell. When interactions within clusters of higher order than pairs (e.g., triplets) are ignored, the excess enthalpy of a supercell can be calculated as a combined effect of all EPI,

*J*

_{ r }(kJ·mol

^{−1}) is the EPI at a distance

*r*,

*f*

_{AB(r)}is the number of pairs of AB-type at a distance

*r*per one exchangeable atom and

*r*

_{max}is the number of dissimilar AB pairs in the supercell.

*r*and

*r*

_{max}in Eq. 8 are understood as integer indexes, with

*r*varying between 1 and

*r*

_{max}. The correspondence between the distance

*r*and the integer index

*r*is established by sorting all the distances in ascending order.

*R*and B

*R*from the enthalpies of the supercell structures with paired substitutional defects of BB-type and AA-type. In the case of (K,Rb)Cl, Eqs. 9 and 10 are used, respectively, for KK and RbRb defect pairs at various distances

*r*,

*H*

_{KK(r)}and ∆

*H*

_{RbRb(r)}are the excess enthalpies of the supercell configurations with double KK and RbRb defects located at a distance

*r*from each other, respectively.

*D*

_{n}is the degeneracy factor (the number of paired defects of a given type that occur due to the periodic boundary conditions). ∆

*H*

_{KK(∞)}and ∆

*H*

_{RbRb(∞)}are considered adjustable parameters and define the excess energy of a hypothetical supercell in which the defect–defect interaction is absent. The parameters ∆

*H*

_{KK(∞)}and ∆

*H*

_{RbRb(∞)}are determined by fitting all DFT energies of the KK and RbRb defect supercell configurations

*k*to Eqs. 11 and 12, respectively. Then,

*J*

_{K(r)}and

*J*

_{Rb(r)}are calculated according to Eqs. 9 and 10.

Here *J* _{K(r)} and *J* _{Rb(r)} are given by Eqs. 9 and 10, respectively. The index *k* denotes a supercell configuration with a KK or RbRb defect at a distance *k* = *r*. This fitting can reproduce the excess energies of all double-defect structures.

*J*

_{ r }characterize the pair ECIs at two extremes along the composition axis. Then, we assume that

*J*

_{ r }at intermediate compositions is a linear combination of

*J*

_{K(r)}and

*J*

_{Rb(r)}in the K- and Rb-limits:

This equation is consistent with the sub-regular behavior of the enthalpy of mixing in the limit of complete disorder and has been used in the study of the NaCl–KCl system by Vinograd and Winkler [10].

When the *J* _{ r } are known, the enthalpy of mixing as a function of the composition and temperature can be obtained using MC simulations. Here, we consider a 10 × 10 × 10 supercell containing 4000 exchangeable atoms (four exchangeable atoms in each unit cell) with periodic boundary conditions. The enthalpy of mixing at a given temperature and composition is calculated with a canonical MC simulation employing the Metropolis algorithm. At each simulation step, a pair of dissimilar exchangeable atoms is chosen randomly and then swapped. When the enthalpy change is ∆*H* ≤ 0, the new configuration is accepted, and when ∆*H* > 0, the new configuration is accepted only if a stochastic generated variable 0 ≤ *ξ* ≤ 1, has *ξ* ≤ exp(−∆*H*/(*kT*)), and it is rejected otherwise. The simulations were performed on a grid of 32 compositions between pure KCl and RbCl and eight temperatures between 73 and 1273 K using a computer program developed by us. In each run, 10^{8} MC steps were performed for the equilibration, and another 10^{8} steps were performed for calculating the average properties.

The enthalpy of mixing determined from MC simulations based on the DFT effective pairwise interactions can also be applied to thermodynamic modeling for the SSAS equilibria. Similar to that for the single-defect method, the MC predicted enthalpy of mixing isotherms will also be used for regressing the parameters *A* _{0} in Eq. 6 and A_{0} and *A* _{1} in Eq. 7. Then, this regular solution parameter or these sub-regular solution parameters will be applied to describe the non-ideality of the solid solution phase in the thermodynamic modeling of the SSAS equilibria in the system KCl + RbCl + H_{2}O.

## 4 Thermodynamic Modeling of SSAS Equilibria

### 4.1 Modeling Methodology of SSAS Equilibria

_{2}O can be determined by minimizing the total Gibbs energy

*G*

^{T}of the system (Eqs. 14–16) with regard to the amount and composition of all phases (aqueous phase and solid solution phase K

_{ x }Rb

_{1−x }Cl) subject to mass and charge balances:

*n*

_{ i }is the mole number of aqueous species

*i*.

*n*

_{KxRb1−xCl}is the mole number of the solid solution K

_{ x }Rb

_{1–x }Cl.

*G*

_{m,i }

^{Θ},

*G*

_{m,KCl}

^{Θ}and

*G*

_{m,RbCl}

^{Θ}are the standard molar Gibbs energies of aqueous species

*i*and solids KCl and RbCl.

*a*

_{ i }is the activity of aqueous species

*i*and is a function of all the

*n*

_{ i }.

*a*

_{KCl}and

*a*

_{RbCl}are the activities of components KCl and RbCl in the solid solution K

_{ x }Rb

_{1–x }Cl, respectively.

*x*is the mole fraction of KCl in the solid solution K

_{ x }Rb

_{1−x }Cl. Equation 16 can be further written as Eq. 17:

*G*

_{m}

^{E}is the molar excess Gibbs energy of the solid solution K

_{ x }Rb

_{1−x }Cl. Equation 6 can be applied to represent

*G*

_{m}

^{E}when the regular solution behavior is followed by K

_{ x }Rb

_{1−x }Cl, while Eq. 7 can be used when sub-regular solution behavior is followed.

An in-house computer program, which we called Institute of Salt Lakes Equilibrium Calculator (ISLEC), written in C language on the Linux platform, was developed by the authors to implement the PSC model, which is advantageous for the representation of the excess thermodynamic properties of a concentrated electrolyte aqueous solution. Multi-component PSC equations without limitations on the number of components are implemented in ISLEC for describing the excess properties of the aqueous phase. The primal–dual interior point filter line search algorithm implemented in the Ipopt software package [60] was called in the ISLEC software to perform Gibbs energy minimization. Here, we use ISLEC to perform all the SSAS equilibrium calculations of the system KCl + RbCl + H_{2}O. Because ISLEC is a command-line based computer program and can only produce numerical results of the solved phase equilibria, the graphical representations of the model results are implemented using separate scientific graphing software, e.g., Origin 8.0.

### 4.2 Model Parameters for SSAS Equilibria Modeling

_{2}O follow our previous studies and are listed in Table 2. The Pitzer–Simonson–Clegg (PSC) model [61, 62] was used to represent the relation between

*a*

_{ i }and all

*n*

_{ i }. The detailed description on the PSC model for the system KCl + RbCl + H

_{2}O can be found in the Appendix. The temperature-dependent binary PSC equation parameters for KCl(aq) and RbCl(aq) were determined in our previous studies [51, 63] and are reproduced here in Table 3. All the PSC mixing parameters

*W*

_{KRbCl},

*U*

_{KRbCl}, and

*Q*

_{1,KRbCl}are set to zero at all temperatures.

Aqueous species or mineral | | |||||||
---|---|---|---|---|---|---|---|---|

| | | | | | | | |

K | 5097.33 | −25943.2 | 49804.1 | −34187.3 | −77.8793 | −1252.516 | 11654.894 | 273–523 |

Rb | 7755.71 | −38369.3 | 71263.5 | −47122.7 | −125.94 | −1816.988 | 17486.448 | 273–523 |

Cl | 29421.3 | −157725 | 316626 | −227316 | −412.404 | −5660.068 | 68302.467 | 273–523 |

H | −90.0669 | 748.166 | −1377.94 | 1055.31 | 3.26118 | −271.203 | −191.886 | 180–523 |

H | 13.44 | 20.2535 | 371.643 | −418.377 | −0.0301987 | −300.209 | 42.01 | 100–273 |

KCl(cr) | 35.4160 | 70.0347 | −91.3823 | 52.5243 | 0.1535 | −447.9278 | 112.3270 | 263–453 |

RbCl(cr) | 49.46 | 3.5098 | 14.3278 | 0 | 0 | −451.8685 | 147.7383 | 253–373 |

Parameter | | |||||
---|---|---|---|---|---|---|

| | | | | | |

| 3.9560903013 (10 | −9.8601244399 (10 | 1.6795521770 (10 | −3.1489849139 (10 | 1.1015107342 (10 | −1.7350430832 (10 |

| 0 | 0 | 0 | 0 | 0 | 0 |

| 1.2987116507 (10 | −3.2812329495 (10 | 5.5990363228 (10 | −1.0553567540 (10 | 3.6745900194 (10 | −5.5544841742 (10 |

| 3.5398179757 (10 | −8.8936745021 (10 | 1.5160440445 (10 | −2.8419620324 (10 | 9.8508754040 (10 | −1.5257774170 (10 |

| −2.4684267843 (10 | 6.1767236620 (10 | −1.0521318451 (10 | 1.9648394849 (10 | −6.791801393 (10 | 1.0700347556 (10 |

| 5.9448776660 (10 | −1.5199277291 (10 | 2.6029424899 (10 | −5.0382050754 (10 | 1.8199193284 (10 | −2.5314443706 (10 |

| 0 | 0 | 0 | 0 | 0 | 0 |

| −1.7103872830 (10 | 4.5270573989 (10 | −7.8100615021 (10 | 1.5789699049 (10 | −5.963220808 (10 | 6.9957353447 (10 |

| −2.1722428837 (10 | 5.7244543837 (10 | −9.8663121027 (10 | 1.9832708704 (10 | −7.446006690 (10 | 8.9265361872 (10 |

| 0 | 0 | 0 | 0 | 0 | 0 |

| 0 | 0 | 0 | 0 | 0 | 0 |

| 0 | 0 | 0 | 0 | 0 | 0 |

| 0 | 0 | 0 | 0 | 0 | 0 |

The regular solution model as indicated by Eq. 6 with parameter *A* _{0} is applied for describing the excess properties of the solid solution phase (K,Rb)Cl. The regular solution parameter *A* _{0} for (K,Rb)Cl has previously been reported by various authors [3, 21, 64, 65, 66]. Sangster and Pelton [64] determined the parameter *A* _{0} = 1500 J·mol^{−1} from experimental solid–liquid equilibrium data in the system KCl–RbCl at a temperature of approximately 1000 K. Ratner and Makarov [65, 66] determined *A* _{0} = 3000 J·mol^{−1} by fitting their determined SSAS equilibrium data in the system KCl + RbCl + H_{2}O at 298.15 K. Glynn [3] determined *a* _{0} = *A* _{0}/*RT* = 1.20, which is equal to *A* _{0} = 3000 J·mol^{−1}, by fitting the SSAS equilibrium data in the system KCl + RbCl + H_{2}O determined by D’Ans and Busch [67] and Merbach and Gonella [58] at 298.15 K. Christov [21] determined *A* _{0} = 4264 J·mol^{−1} and *A* _{0} = 3325 J·mol^{−1} by fitting the SSAS equilibrium experimental data at 298.15 K reported by D’Ans and Busch [67] and Ratner and Makarov [65, 66], respectively. Comprehensively considering the available values of *A* _{0}, we suppose that *A* _{0} = 3000 J·mol^{−1} at *T* = 298.15 K may be more reliable. As an approximation, the parameter *A* _{0} is seen as temperature independent in the limited temperature range of 273.15 K to 373.15 K concerned in the present study.

### 4.3 Graphical Representation of SSAS Equilibria

The Gibbs triangular phase diagram, Roozeboom diagram and Lippmann diagram were used as graphical representations of the experimental and model-calculated SSAS equilibria in the studied system KCl + RbCl + H_{2}O.

The Gibbs triangular phase diagram is the most commonly used graphical representation of ternary phase equilibria. Here, the right triangle form of the Gibbs diagram with mass percentage coordinates of the solutes KCl and RbCl, indicated as 100*w*, was used to represent the liquid composition curve for testing the consistency of the available experimental data and model-calculated results.

The Roozeboom diagram, which is advantageous for the representation of SSAS equilibria, was applied to represent the partitioning of Rb between the aqueous and solid phases. The coordinates of the Roozeboom diagram are the mole fraction of Rb, i.e., *x* _{Rb}(ss) = *n* _{Rb}(ss)/{*n* _{Rb}(ss) + *n* _{K}(ss)} and *x* _{Rb}(aq) = *n* _{Rb}(aq)/{*n* _{Rb}(aq) + *n* _{K}(aq)}, where ss denotes the solid solution phase, aq denotes the aqueous solution phase and *n* _{Rb} and *n* _{K} denote the mole numbers of Rb and K in the solid solution and aqueous solution phases. This type of diagram representation of the SSAS equilibria can be applied to reflect the composition information of the aqueous and solid phases simultaneously. However, in the Roozeboom diagram, only *x* _{Rb}(aq) is presented instead of the complete chemical composition of the aqueous phase. That is, the real concentration of the solute in the aqueous phase is not reflected in the Roozeboom diagram.

In addition to the Gibbs and Roozeboom diagrams, the Lippmann diagram was introduced to represent the SSAS equilibria in the studied system KCl + RbCl + H_{2}O. Following Lippmann [26] and Glynn and Reardon [2], we define the total solubility product, \( \varSigma \varPi \), of a solid solution A_{ x }B_{1–x }C as ({A^{+}}+{B^{+}}){C^{−}}, where {A^{+}}, {B^{+}} and {C^{−}} are the activities of aqueous species A^{+}, B^{+} and C^{−} in the aqueous phase. In the Lippmann diagram, \( \varSigma \varPi \) or its logarithm is plotted as the y-axis against two superimposed x-axes: (i) the activity fraction of the substituting ions in the aqueous solution, representing the solutus function \(x_{\text{aq},{\text{B}^+}}\) ≡ {B^{+}}/({A^{+}} + {B^{+}}), and (ii) the mole fraction of the end member in the solid phase *x* _{BC}, representing the solidus function. In this study, we follow the original definitions in the Lippmann diagram representation in spite of some variations that have also been used in literature.

*x*

_{AC}and

*x*

_{BC}are the mole fractions of end members AC and BC, respectively, in the solid solution phase,

*γ*

_{AC}and

*γ*

_{BC}are the activity coefficients of end members AC and BC, respectively, in the solid solution phase, and

*K*

_{AC}and

*K*

_{BC}are the solubility products of pure solids AC and BC, respectively. \( \varSigma \varPi_{\text{eq}} \) refers to the value of \( \varSigma \varPi \) as specifically defined at thermodynamic equilibrium.

*γ*

_{AC},

*γ*

_{BC},

*K*

_{AC}and

*K*

_{BC}have the same meaning as in Eq. 18, and

*x*

_{aq,A+}and

*x*

_{aq,A+}are the activity fractions of the substituting ions in the aqueous solution.

^{+}}

_{eq}, {B

^{+}}

_{eq}and {C

^{−}}

_{eq}are the activities of aqueous species A

^{+}, B

^{+}and C

^{−}in the aqueous phase in equilibrium with a solid solution.

It is notable that Eq. 18 is a function of the properties of only the solid solution phase, and Eq. 20 is a function of the properties of only the aqueous solution phase. In Eq. 19, the properties of both the solid solution and aqueous solution phases are involved. Theoretically, at the thermodynamic equilibrium state, the values of \( \varSigma \varPi_{\text{eq}} \) calculated using Eqs. 18–20 should be strictly equal to each other. This can be assured when modeling approaches are used, but it may not be assured when experimental data are plotted due to the possible inaccuracy of these data. To plot experimental data on the Lippmann diagram, we use Eq. 19 to calculate \( \varSigma \varPi_{\text{eq}} \) from experimental data all the time in the present study because it simultaneously considers the thermodynamic properties of the solid solution and aqueous solution phases and is not excessively sensitive to the uncertainty of the experimental data. This process is consistent with the treatment of Glynn et al. [2, 4].

## 5 Results and Discussion

### 5.1 Experimental Results and Discussion

_{2}O determined in this study at

*T*= 298 K are listed in Table 4. The composition of the solid phase,

*x*

_{RbCl,}was determined using various methods, and the results are listed in Table 5. Figure 2 shows those experimental data in a Gibbs triangular phase diagram. For each phase equilibrium experiment, at the equilibrium state, the determined liquid point, system point, wet residual point and the solid point determined from Vegard’s rule can be connected using a straight line, indicating that the liquid composition determined by the chemical analysis and the solid composition determined by Vegard’s rule are consistent with each other. Figure 3 shows the differences among the

*x*

_{RbCl}determined from the lattice parameter Vegard’s rule, lattice volume Vegard’s rule, Schreinemakers wet residual method and system point method, indicating that the compositions of the solid phase determined from various methods are in good agreement. Moreover, the values of

*x*

_{RbCl}determined from the Schreinemakers wet residual method are in agreement with those determined from the lattice parameter Vegard’s rule within an absolute deviation of 1.5%. The variation of the XRD patterns (the two strongest peaks) of the solids with the chemical composition are plotted in Fig. 4, compared with those of pure KCl(cr) and RbCl(cr), and the determined lattice parameter and volume are listed in Table 6.

Solid–liquid phase equilibria data of the KCl + RbCl + H_{2}O system at *T* = 298 K determined in this study

No. | System composition/(100 | Solution composition/(100 | Wet residual composition/(100 | ||||||
---|---|---|---|---|---|---|---|---|---|

KCl | RbCl | H | KCl | RbCl | H | KCl | RbCl | H | |

1 | 6.06 | 50.36 | 43.58 | 4.54 | 42.57 | 52.89 | 7.83 | 59.74 | 32.43 |

2 | 11.79 | 42.17 | 46.04 | 7.97 | 37.25 | 54.78 | 18.29 | 47.43 | 34.28 |

3 | 16.20 | 34.85 | 48.95 | 10.14 | 33.17 | 56.69 | 27.20 | 36.23 | 36.57 |

4 | 21.44 | 26.55 | 52.01 | 12.99 | 27.41 | 59.60 | 35.72 | 26.24 | 38.04 |

5 | 27.16 | 17.46 | 55.38 | 16.95 | 18.78 | 64.27 | 41.83 | 16.31 | 41.86 |

6 | 33.93 | 8.36 | 57.71 | 21.96 | 9.37 | 68.67 | 51.43 | 6.8 | 41.77 |

Composition of the solid phases determined using different methods

No. | Composition of solid phase | |||
---|---|---|---|---|

| | | | |

1 | 0.802 | 0.804 | 0.818 | 0.811 |

2 | 0.568 | 0.528 | 0.538 | 0.527 |

3 | 0.340 | 0.307 | 0.314 | 0.304 |

4 | 0.138 | 0.164 | 0.182 | 0.175 |

5 | 0.059 | 0.075 | 0.078 | 0.075 |

6 | 0.019 | 0.018 | 0.010 | 0.009 |

Lattice parameters and volumes determined from the powder XRD pattern and cell refinement of the solid solution samples^{a}

No. | Lattice parameter | Lattice volume |
---|---|---|

1 | 6.53939 | 279.6470 |

2 | 6.45703 | 269.2145 |

3 | 6.39077 | 261.0114 |

4 | 6.35187 | 256.2741 |

5 | 6.32120 | 252.5798 |

6 | 6.30104 | 250.1708 |

_{2}O from various sources are illustrated in Fig. 6 for comparison. In the case of the liquidus, the data from different sources agree well with each other, while most of the data for the composition of the solid phase do not agree with each other, and significant differences can be observed for the cases with approximately the same liquid point. Hence, for a SSAS equilibrium system, the consistency of data from different sources cannot be ensured solely from the good agreement among the liquid points; the consistency of the solid compositions is also of importance. Although the data reported by D’Ans and Busch [67] show good agreement with the present data in terms of the liquidus, significant differences, for absolute deviations of approximately 10% in the mass percentage of KCl in the solid solution, can be found in terms of the solid compositions corresponding to approximately the same liquid points. As a whole, the experimental data of Merbach and Gonella [58] agree with those of the present study for both the liquid and solid points. In the KCl-rich region, the data of Yu et al. [68] agree with the present study for both the liquid and solid points, while in the RbCl-rich corner, significant differences can be found.

Even though the compositions of the solid phase reported by different sources are all determined from the chemical compositions of the liquid point and wet residual point, significant differences exist among these data. This indicates that the Schreinemakers’ wet residues method cannot always ensure that reliable solid solution compositions are obtained when SSAS equilibria are studied. Due to different experimental skill levels of researchers and wet residual sampling and post-processing details, significant differences exist among the reported solid solution compositions in equilibrium with a single liquid point, although they all used the Schreinemakers’ wet residues method. Compared with the Schreinemakers’ wet residues method, however, the physical method based on the XRD technique and Vegard’s rule may be a simpler, faster, more reliable and researcher-independent alternative. However, it should be noted that not all solid solutions follow Vegard’s rule. The XRD quantitative approach is valid only if Vegard’s rule is already known to be followed by a specific solid solution phase or the relationship between the lattice parameters and the composition of the solid solution is well known. Additionally, because the lattice parameters are insensitive to slight changes in the solid solution composition, the method will lose its accuracy when the mole fraction of any component in the solid solution is less than 1%.

### 5.2 Atomistic Simulation Results and Discussion

The unit cell parameters determined from the DFT geometry optimization are *a* = *b* = *c* = 6.332 Å, *α* = *β* = *γ* = 90° (*V* = 253.88 Å^{3}) for KCl(cr) and *a* = *b* = *c* = 6.681 Å, *α* = *β* = *γ* = 90° (*V* = 298.21 Å^{3}) for RbCl(cr). Similar to other defected structures, the geometry optimization of the two pure substances was also performed in a 2 × 2 × 2 supercell. The unit cell parameters were seen as half of the lattice parameters of the supercell. Those values agree with the experimental values, *a* = *b* = *c* = 6.299 Å for KCl and *a* = *b* = *c* = 6.581 Å for RbCl, within an uncertainty of 1.5%.

*H*

_{m,mix}(listed in Table 7) of all the defected structures in the 2 × 2 × 2 supercell of the (K,Rb)Cl solid solution were calculated using Eq. 4 relative to the 2 × 2 × 2 supercells of pure KCl and RbCl. The regular solution model, as indicated by Eq. 6, was applied to represent the two

*H*

_{m,mix}because only slight differences can be found between the molar excess enthalpies of the two single-defect structures. For the (K,Rb)Cl solid solution, a regular solution parameter

*A*

_{0}= 4918 J·mol

^{−1}was ultimately determined by the single-defect method.

Molar enthalpy of mixing (*H* _{m,mix}) of single- and double-defect structures in the 2 × 2 × 2 supercell of the (K,Rb)Cl solid solution calculated using DFT

Type | |
---|---|

Single defects | |

Rb in KCl | 159.113 |

K in RbCl | 138.643 |

| | D | |
---|---|---|---|

Double defects | |||

RbRb defects in KCl | |||

1 | 4.454 | 2 | 294.474 |

2 | 6.299 | 2 | 362.723 |

3 | 7.714 | 2 | 301.907 |

4 | 8.908 | 4 | 316.178 |

5 | 10.91 | 8 | 301.930 |

KK defects in RbCl | |||

1 | 4.454 | 2 | 337.233 |

2 | 6.299 | 2 | 296.512 |

3 | 7.714 | 2 | 256.667 |

4 | 8.908 | 4 | 272.901 |

5 | 10.91 | 8 | 263.865 |

*J*

_{ r }, determined from the DFT energies of the double-defected supercell structures are listed in Table 8. Figure 7 shows the

*J*

_{ r }converging to zero with increasing distance between the defects, which indicates that the effective pairwise interactions can be omitted when the distance between the defects is greater than 10.910 Å. Similar limiting distances, 9.734 Å for (Na,K)Cl, 12.629 Å for (Ba,Ra)SO

_{4}and 12.156 Å for (Ca,Sr)CO

_{3}, were recognized by Vinograd et al. [8, 10, 42]. The enthalpy of mixing isotherms calculated from Monte Carlo simulations with the DFT-based

*J*

_{ r }were plotted in Fig. 8. Clearly, the enthalpy of mixing increases with the temperature, but the magnitude of the increase is relatively small when the temperature is above 273 K. The regular solution parameter

*A*

_{0}fitted from the MC-generated enthalpy of mixing isotherms is 5103, 5372 and 5529 J·mol

^{−1}at 273, 373 and 573 K, respectively. These values are similar to that determined using the single-defect method. However, compared with the regular solution parameter range from 3000 to 4000 J·mol

^{−1}regressed from the SSAS equilibrium data in the KCl + RbCl + H

_{2}O system at 298.15 K, these values are significantly larger. The poor agreement may imply the existence of a non-negligible excess entropy of mixing. A similar conclusion was drawn when the NaCl–KCl system was studied by Vinograd and Winkler [8]. Further improvement of the simulation may require the use of a temperature-dependent

*J*

_{ r }.

Excess enthalpies (Δ*H*) of the 2 × 2 × 2 supercell structures of the K_{ x }Rb_{1−x }Cl solid solution with paired KK and RbRb defects and the corresponding pairwise interaction energies (*J* _{ r })

| | Δ | |
---|---|---|---|

KK defects | |||

1 | 4.454 | 10.737 | 0.332 |

2 | 6.299 | 9.441 | −0.754 |

3 | 7.714 | 8.172 | 0.214 |

4 | 8.908 | 8.689 | −0.007 |

5 | 10.91 | 8.401 | 0.053 |

| | | |
---|---|---|---|

RbRb defects | |||

1 | 4.454 | 9.376 | −0.728 |

2 | 6.299 | 11.549 | −0.079 |

3 | 7.714 | 9.612 | 0.555 |

4 | 8.908 | 10.067 | 0.148 |

5 | 10.91 | 9.613 | 0.110 |

To investigate whether we can obtain satisfactory predictions on the SSAS equilibrium of the KCl + RbCl + H_{2}O system using the DFT-predicted mixing thermodynamic properties of the (K,Rb)Cl solid solution, we used the regular solution parameter *A* _{0} = 4918 J·mol^{−1} predicted with SDM, which is similar to those predicted with DDM in the temperature range of 273–573 K. The SSAS equilibrium modeling results with DFT mixing thermodynamics will be discussed in Sect. 5.3.

### 5.3 Thermodynamic Modeling Results and Discussion

_{2}O at

*T*= 298.15 K compared with experimental values reported in the literature [69]. Good agreement can be found between the model predictions and experimental values, indicating that it is reasonable to set all the mixing parameters in the PSC activity coefficient equations to zero for the system of KCl + RbCl + H

_{2}O.

_{2}O at 298.15, 323.15 and 373.15 K, represented by a Gibbs phase diagram, Roozeboom diagram and Lippmann diagram, respectively. The model-predicted liquid curves using the regular solution model assumption for the solid solution phase (K,Rb)Cl agree well with the available experimental data at 298.15 K (see Fig. 10a), 323.15 K (see Fig. 11a) and 373.15 K (see Fig. 12a), no matter whether the value of the dimensionless parameter used is

*A*

_{0}= 3000 J·mol

^{−1}, as reported in the literature, or

*A*

_{0}= 4918 J·mol

^{−1}, as predicted by the present DFT calculations. The model-predicted liquid curves with the ideal solution model assumption for (K,Rb)Cl are clearly lower than the experimental points at all three temperatures.

However, when the information of the liquidus and solidus is presented simultaneously, the model results using the ideal solution model and regular solution model with an *A* _{0} value of 4918 J·mol^{−1} are not in agreement with the experimental data at any temperature, as indicated in Fig. 10b–d, Fig. 11b–d and Fig. 12b–d. At 323.15 K, the model with a regular solution parameter *A* _{0} = 3000 J·mol^{−1} gives the best predictions compared with the ideal solution model and the regular solution model with *A* _{0} = 4918 J·mol^{−1}, while at 373.15 K, none of the model results are consistent with the experimental data, no matter whether the ideal solution model or regular solution model with *A* _{0} = 3000 J·mol^{−1} or 4918 J·mol^{−1} is used for representing the excess Gibbs energy of the solid solution phase (K,Rb)Cl (see Fig. 12b–d). It is notable that, as shown in Fig. 12b, c, the model-predicted results using the regular model with *A* _{0} = 3000 J·mol^{−1} are still consistent with the experimental values at both sides enriched with KCl and RbCl at *T* = 373.15 K, while in the middle range, significant differences can be observed. The poor agreement between the model results and the experimental data at 373.15 K may result from the improper model parameters or unreliable experimental data, but we tend to believe it results from the latter. Because there are no other experimental data at 373.15 K that can be found in the literature except for the one set used here, the reliability of the experimental data, especially for the compositions of the solid phase, cannot be evaluated critically. It is clear that the experimental data at 298.15 K and 323.15 K can be well represented with the present model, in which the regular model with parameter *A* _{0} = 3000 J·mol^{−1} is applied to represent the mixing thermodynamics of the (K,Rb)Cl solid solution. Hence, we suppose that the assumption that *A* _{0} = 3000 J·mol^{−1} and is constant in the studied temperature range is reasonable. However, new experimental data on the SSAS equilibria in the ternary system KCl + RbCl + H_{2}O at temperatures other than 298.15 K are still necessary to verify the reliability of the present thermodynamic model.

_{2}O system at the KCl-rich side, which is of particular importance in natural and industrial processes, the distribution coefficients {

*D*

_{Rb}= (

*n*

_{Rb}/

*n*

_{K})

_{ss}/(

*n*

_{Rb}/

*n*

_{K})

_{aq}}} of the minor Rb between the KCl + RbCl + H

_{2}O system aqueous solution and (K,Rb)Cl solid solution at various temperatures were calculated and compared with experimental values reported in the literature [19, 73]. The available experimental values of

*D*

_{Rb}are scattered as shown in Fig. 13, but the model results represent the average values at each temperature roughly in the temperature range of 273.15–373.15 K. This indicates that the present model is relatively reliable.

## 6 Conclusions

- 1.
The solid–liquid phase equilibria in the ternary KCl + RbCl + H

_{2}O system were carefully redetermined at 298.15 K. For every equilibrium experiment, two approaches, i.e., the Vegard approach and Schreinemakers’ wet residues approach, was used to determine the composition of the solid phase. The values determined from the two approaches are in good agreement, and the validity of the Vegard approach for the determination of the composition of the solid solution phase (K,Rb)Cl is confirmed. The Vegard approach will be further applied to multicomponent systems, e.g., NaCl + KCl + RbCl + H_{2}O and KCl + RbCl + MgCl_{2}+ H_{2}O, for the determination of the compositions of (K,Rb)Cl when it coexists with other solids. The liquidus determined in the present study is in accordance with those reported in the literature, but differences may be found when these SSAS equilibrium experimental data are compared with each other using the couples of liquid and solid points. - 2.
The validity of the atomistic simulation methods for directly predicting the mixing properties of the (K,Rb)Cl solid solution was studied. The DFT calculation and DFT-based MC simulations give nearly the same regular solution parameter for the (K,Rb)Cl solid solution. However, the SSAS equilibrium in the KCl + RbCl + H

_{2}O system cannot be represented accurately when those regular solution parameters are used. - 3.
A temperature-dependent thermodynamic model for the system of KCl + RbCl + H

_{2}O has been developed based on our previously determined binary model parameters for the KCl + H_{2}O and RbCl + H_{2}O systems and a regular solution model with a dimensionless parameter*A*_{0}= 3000 J·mol^{−1}for (K,Rb)Cl. In the model, the mixing parameters in the PSC equations are unnecessary. The reliability of the present model has been verified by comparing the model-predicted values with the complete SSAS equilibrium properties in the KCl + RbCl + H_{2}O system at 298.15 and 323.15 K. The present model is also valid to predict the partition coefficients of Rb between the aqueous and solid phases in the temperature range of 273–373 K, even when minor amounts of Rb are dissolved in an aqueous solution of KCl. The present model of the system KCl + RbCl + H_{2}O is an important component of our final goal to develop a comprehensive thermodynamic model for the multicomponent system Na, K, Rb, Mg//Cl−H_{2}O.

## Notes

### Acknowledgements

One of the author (D. Gao) would like to acknowledge the National Natural Science Foundation of China–Qinghai Government United Project on Qaidam Salt Lake Chemical Engineering Science (U1407131) for financial supporting. Prof. Bo Yang is acknowledged for collecting the XRD data. The National Supercomputing Center in Shenzhen is acknowledged for providing the computation resources. We would like to thank Dr. D.A. Kulik for his suggestions on the manuscript improvement, discussions on correct usage of the Lippmann diagram, and some constructive comments on the atomistic simulation and thermodynamic modeling parts. We are also very grateful to the reviewer, Prof. M. Prieto, for his comments on the experimental part, especially for the discussions on the homogeneity characterization and composition analysis of solid solution.

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