1 Introduction

The studies of phase equilibria in multicomponent salt–water systems at different temperatures are fundamental for the development and utilization of brine resources. Bromide rich brines are widely distributed in the Qinghai–Tibet plateau and the Sichuan basin in China. In recent decades, some scholars have investigated the phase equilibria of parts of the salt–water systems [1, 2]. In particular, the underground brines in the western Sichuan basin contain very abundant liquid mineral resources. The contents of boron, potassium, and bromine in these brines are far above the grades necessary for industrial utilization [3, 4]. The underground brines have stable chemical properties, and have high concentrations of potassium, sodium, boron, bromine, as well as accompanying lithium, strontium, magnesium, calcium, and many other useful components.

Underground brines are very complex salt–water systems, whose formation often involve solid–liquid equilibria. Furthermore, the phase diagrams at different temperatures play an important role in exploiting the brine resources and understanding the geochemical behavior of these brine–mineral systems. Therefore, it is necessary and urgent to investigate the thermodynamics (especially the phase equilibrium properties) of Br-rich brine–mineral systems at multiple temperatures.

A number of experimental and modelling studies on brine–salt mineral system have been carried out in recent decades. Aiming at the seawater system, Harvie et al. [5–8] carried out research into the prediction of mineral solubilities in natural waters of the Na–K–Mg–Ca–H–Cl–SO4–OH–HCO3–CO3–CO2–H2O systems from zero to high concentration and as a function of temperature using Pitzer’s equation. Felmy and Weare [9] predicted the borate mineral equilibria in Searles salt lake in the USA. In view of the abundant bromide resource in potash salt deposits, the phase equilibria and phase diagrams of the ternary KCl–KBr–H2O system was determined at 298, 313, 333, 323 and 373 K, and the LiI–LiBr–H2O and NH4Br–LiBr–H2O systems at 298 K [10–13]. In our previous work, we reported the phase diagrams of some bromide-rich ternary and quaternary systems, i.e., K2B4O7–KBr–H2O and Na2B4O7–NaBr–H2O at 298 K [14, 15], Na–K–Br–SO4–H2O at 323 K [16], NaCl–NaBr–Na2SO4–H2O at 348 K [17], Na2B4O7–NaBr–Na2SO4–H2O at 348 K [18], and KCl–KBr–K2SO4 at 323 K [19].

Recently, Christov reported predictions of the solubilities of bromide salts and solution behavior in solid–liquid equilibrium systems [20–24]. The thermodynamic modeling was carried out to high concentration and temperature with Pitzer’s equation. Due to the lack of solubility data involving bromide salts at different temperatures, the calculations were mainly focused on binary and ternary systems such as CaBr2–H2O, MgBr2–H2O, NaBr–H2O, KBr–H2O, NaBr–KBr–H2O, NaBr–Na2SO4–H2O, and KBr–K2SO4–H2O [20–24].

Therefore, this paper has three objectives: (1) measure the solubilities of salts in the aqueous solutions of the quaternary systems NaBr–KBr–CaBr2–H2O and NaBr–KBr–CaBr2–H2O at 348.15 K, identify the equilibrium solid phases and construct the experimental phase diagrams and water content diagrams of the two quaternary systems; (2) use Pitzer’s equation to fit the dissolution equilibrium constant of salts; and (3) use these thermodynamic data to develop a thermodynamic model for the prediction of bromide salt solubilities in the two quaternary systems.

2 Experimental Sections

2.1 Reagents and Instruments

Distilled water with conductivity less than 10−4 S·m−1 and pH = 6.6 was used for the phase equilibrium experiments and chemical analysis of samples. A HZ–9613Y type thermostated shaker with a precision ±0.1 K was used in the solid–liquid equilibrium measurements. NaBr, 99.0 % pure, was from Tianjin Zhiyuan Chemical Reagent Factory, China; MgBr2·6H2O and CaBr2·2H2O, 98 % pure, were from Shanghai Xinbao Fine Chemical Plant, China; KBr, 99.0 % pure, was from the Chengdu area of the industrial development zone, Xindu Mulan, China.

2.2 Experimental Methods

The experiments were conducted by the method of isothermal solution saturation. Samples of the quaternary systems were prepared by gradual addition of the third salt to the saturated solution of the ternary subsystems at 348.15 K. The solid reagents and water were prepared in 100 mL glass bottles. Then, all bottles were sealed and placed in the thermostated shaker. The solid–liquid mixtures in the bottles were agitated for 1 week to accelerate equilibration. Experimental results show that equilibrium can be attained in 10–15 days under continuous shaking, and the time for the clarification of an aqueous solution is about 4–6 days. The clarified solutions were taken out periodically for chemical analysis. If the solution composition did not change, the system can be considered to have reached thermodynamic equilibrium. In this state, the solution composition should represent that of the saturated solution. Liquid samples were then taken for chemical analysis.

2.3 Analytical Methods

The concentration of K+ was obtained by a gravimetric method using sodium tetraphenylborate (uncertainty: 0.5 %). The concentration of Mg2+ was determined by titration with an EDTA standard solution in the presence of the indicator eriochrome black T; an ammonium chloride buffer solution was used to maintain the high pH (10–11) of solutions in the complexometric titration process (uncertainty, 0.3 %). The concentration of Ca2+ was determined by titration with a standard EDTA solution in the presence of the indicator eriochrome K–B, while a sodium hydroxide solution was used for maintaining the high pH (11–12) of solutions in the complexometric titration (uncertainty: 0.5 %). The concentration of Br− was measured by titration with silver nitrate solution (the Mohr method, uncertainty: ±0.3 %). The concentration of Na+ was evaluated according to the charge balance of ions. The details of the above analytical methods can be found in the literature [25].

3 Results and Discussions

3.1 The KBr–NaBr–MgBr2–H2O System

The experimental solubilities of salts in the quaternary system KBr–NaBr–MgBr2–H2O at 348.15 K are given in Table 1. According to the solubilities, the corresponding dry salt compositions were calculated and are also tabulated in Table 1. Based on the dry salt composition, the phase diagram and water content diagram are presented in Figs. 1 and 2, respectively.

Table 1 Solubilities in the quaternary system NaBr–KBr–MgBr2–H2O at 348.15 K
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Fig. 1
figure 1

Dry-salt solubility diagram of the quaternary system NaBr–KBr–MgBr2–H2O at 348.15 K

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Fig. 2
figure 2

Water contents of saturated solutions in the quaternary system NaBr–KBr–MgBr2–H2O at 348.15 K

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It can be seen from Table 1 and Fig. 1 that the double salt (KBr·MgBr2·6H2O) is formed in the quaternary system. The phase diagram (Fig. 1) is comprised of five univariant curves (A1E1, E1F1, B1E1, C1F1 and D1F1) and two co-saturated points (E1 and F1). The four crystallization fields are MgBr2·6H2O, KBr·MgBr2·6H2O, NaBr and KBr, respectively. The crystallization field of MgBr2·6H2O is smaller, which indicates that its solubility is greater in this system. The experimental results show that MgBr2 has a strong salting-out effect on KBr and NaBr.

Figure 2 shows the water content diagram of the quaternary system KBr–NaBr–MgBr2–H2O at 348.15 K and the abscissa is the Jänecke index of J (KBr). Figure 2 shows that the water content increases at the univariant curves D1F1, and decrease at the univariant curves A1E1 with an increase of the Jänecke index values of J (KBr).

3.2 The KBr–NaBr–CaBr2–H2O System

The measured values of salt solubilities of the quaternary system KBr–NaBr–CaBr2–H2O at 348.15 K are presented in Table 2, where ion concentrations are expressed in mass fractions w, J is the Janeäcke index, with J(KBr) + J(NaBr) + J(CaBr2) = 100 g. Based on the data in Table 2, a stable equilibrium phase diagram of the system at 348.15 K is given in Fig. 3.

Table 2 Solubilities in the quaternary system NaBr–KBr–CaBr2–H2O at 348.15 K
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Fig. 3
figure 3

Dry-salt solubility diagram of quaternary system NaBr–KBr–CaBr2–H2O at 348.15 K

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The quaternary system KBr–NaBr–CaBr2–H2O at 348.15 K has no complex salt or solid solution. There is one invariant point, three univariant curves, and three regions of crystallization in this system. The three crystallization fields corresponding to KBr, NaBr and CaBr2·2H2O. The crystallization area of potassium bromide (KBr) is larger than that of other salts. This means that KBr has a lower solubility than the other salts in the quaternary NaBr–KBr–CaBr2–H2O system.

Three univariant curves are A2E2, B2E2, and C2E2. Point E2 is the invariant point for the system KBr–NaBr–CaBr2–H2O at 348.15 K, which is saturated with KBr, NaBr, and CaBr2·2H2O, and the composition of the corresponding liquid phase is w (KBr) = 0.0421, w (NaBr) = 0.0929, and w (CaBr2) = 0.5708.

Figure 4 shows the water content diagram of the quaternary system KBr–NaBr–CaBr2–H2O at 348.15 K, and the abscissa is the Jänecke index values of KBr. Figure 4 shows that the water content increases at the univariant curves E2C2, and remains unchanged at the univariant curves A2E2 and B2E2 with an increase of the Jänecke index values of J (KBr). It reaches its biggest value at the point C2.

Fig. 4
figure 4

Water contents of saturated solutions in the quaternary system NaBr–KBr–CaBr2–H2O at 348.15 K

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4 Solubility Predictions

4.1 Model Parametrization

Pabalan [26] proposed a high temperature thermodynamic model for the concentrated electrolyte solution system Na–K–Mg–Ca–SO4–OH–H2O over a temperature range (from 273.15 to 473.15 K). In order to obtain thermodynamic properties and parameters of the ternary solutions, such as osmotic coefficients ϕ, binary and ternary ionic interaction parameters, and dissolution equilibrium constant K, the following equations were applied to the phase equilibria and chemical equilibria of a given solution [27]:

$$ {\text{Par}}(T) = a_{ 1} { + }a_{ 2} T{ + }a_{ 5} T^{ - 1} + a_{6} lnT $$
(1)
$$ K({\text{KBr}}) = \gamma_{ \pm }^{2} m^{2} $$
(2)
$$ K({\text{NaBr}}) = \gamma_{ \pm }^{2} m^{2} $$
(3)
$$ K({\text{MgBr}}_{2} \cdot 6{\text{H}}_{2} {\text{O}}) = 4\gamma_{ \pm }^{3} m^{3} a_{\text{w}}^{6} $$
(4)
$$ K({\text{CaBr}}_{2} \cdot 2{\text{H}}_{2} {\text{O}}) = 4\gamma_{ \pm }^{3} m^{3} a_{\text{w}}^{2} $$
(5)
$$K(\rm{KBr}\cdot \rm{MgBr}_2\cdot 6\rm{H_2O})= m_{K^+}\cdot \gamma_{K^+}\cdot m_{Mg^{2+}}\cdot (m_{Br^{-}}\cdot \gamma_{Br^{-}})^{3}\cdot a_w^6$$
(6)

where Par(T) refers to temperature functions for the model parameters as described in Møller (1988); the temperature T is in Kelvin. The Debye–Hückel constant \( A^{\phi } \) was taken from the literature [28]. The parameters \( a_{1} ,\;a_{2} ,\;a_{5} \) and \( a_{6} \) are based on the results of Christov [20]. In Eqs. 2–5, \( \gamma_{i} \) and m i are the activity coefficient and molality of solute species i, respectively.

4.2 Solubility Calculation

Based on Pitzer’s ion-interaction model for aqueous electrolyte solutions and the corresponding temperature-dependent parameter expressions, the dissolution equilibrium constant (K) of solid salts (MgBr2·6H2O, KBr·MgBr2·6H2O, NaBr, KBr and CaBr2·2H2O) were fitted by multiple linear regression method. The constants in Eq. 1 for the binary and ternary interaction parameters are taken from the literature [20] (see Table 3). Table 4 shows the calculated dissolution equilibrium constant (ln K) values at 348.15 K.

Table 3 Binary and ternary interaction parameters in Eq. 1
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Table 4 Dissolution equilibrium constant of salts (K) at 348.15 K calculated in this work
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Harvie and Weare [6] showed that the Pitzer approach could be extended to accurately calculate salt solubilities in complex brines. The solubilities of salts in two quaternary systems at 348.15 K were calculated with the Pitzer model. As shown in Figs. 5 and 6, the experimental and predicted phase diagrams (dashed line) of the two quaternary systems at 348.15 K are in general agreement. Tables 5 and 6 give a comparison of the calculated and experimental solubilities at the invariant points of the two quaternary systems and the saturated solution compositions of the relevant quaternary subsystems, respectively. As can be seen from Figs. 5 and 6, there are some deviations between the calculated and experimental solubilities. The sources of deviations include (1) experimental uncertainty, and (2) uncertainties of the ion-interaction parameters of Pitzer’s equation due to the regression and the experimental data used in the regression, although the regression equation is applicable only in a certain temperature range (the accuracy of the ion-interaction model requires further validation at high temperatures and high concentrations). Overall, the above results indicate that the model used in this work is capable of a reasonable prediction of the salt solubilities for the quaternary systems in question.

Fig. 5
figure 5

Comparison of calculated and experimental solubilities of salts in the KBr–NaBr–MgBr2–H2O system at 348.15 K: experimental (filled circle); calculated (filled triangle)

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Fig. 6
figure 6

Comparison of calculated and experimental solubilities of salts in the quaternary system KBr–NaBr–CaBr2–H2O at 348.15 K: experimental (filled circle); calculated (filled triangle)

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Table 5 Comparison of experimental and calculated invariant points of the quaternary system KBr–NaBr–MgBr2–H2O and its binary subsystems at 348.15 K
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Table 6 Comparison of experimental and calculated invariant points of the quaternary system KBr–NaBr–CaBr2–H2O and its binary subsystems at 348.15 K
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5 Conclusion

The solubilities of salts in the systems KBr–NaBr–MgBr2–H2O and KBr–NaBr–CaBr2–H2O at 348.15 K were determined with the method of isothermal solution saturation. According to the experimental data, the stable equilibrium phase diagrams and water content diagrams of the two quaternary systems were constructed. Based on Pitzer’s model and temperature-dependent parameter expressions, the dissolution equilibrium constant of the salts are represented. The solubilities of salts in the quaternary systems were predicted at 348.15 K. The results indicate that the Pitzer model based on the temperature-dependent parameter expressions is applicable for the prediction of solubilities in the quaternary systems.