Solid–Liquid Phase Equilibria of the Aqueous Quaternary System Rb⁺, Cs⁺, Mg²⁺//SO₄²⁻ - H₂O at T = 323.2 K

Abstract

Sulfate-type salt lakes constitute over half of the total salt lakes in China and are rich in rare elements, such as rubidium and cesium. However, the complex interactions between ions make the separation and extraction process quite challenging. To address this, phase equilibrium studies were conducted on the sulfate system containing rubidium, cesium, and magnesium. Specifically, the phase equilibria of the aqueous quaternary system Rb⁺, Cs⁺, Mg²⁺//SO₄²⁻ - H₂O at 323.2 K were investigated using the isothermal dissolution method. The solubility, density, and refractive index of the system were experimentally measured. The results indicate that the system at 323.2 K belongs to a complex type with the formation of one solid solution (Rb, Cs)₂SO₄ and two double salts (Rb₂SO₄·MgSO₄·6H₂O, Cs₂SO₄·MgSO₄·6H₂O). The corresponding phase diagram consists of four quaternary invariant points, nine univariate curves, and six crystallization regions. Among these, the crystalline region for Cs₂SO₄·MgSO₄·6H₂O is the largest, while that for the single salt Cs₂SO₄ is the smallest. Moreover, the crystalline regions for the double salt and solid solutions are significantly larger than those for the single salt, highlighting the difficulty in separation of valuable single salts. A comparison of multi-temperature phase diagrams from 298.2 K to 323.2 K reveals that the crystalline form of MgSO₄ changes from MgSO₄·7H₂O (298.2 K) to MgSO₄·6H₂O (323.2 K). As the temperature increases, the phase regions for Rb₂SO₄, Cs₂SO₄, (Rb, Cs)₂SO₄, and Cs₂SO₄·MgSO₄·6H₂O expand, while the phase region of Rb₂SO₄·MgSO₄·6H₂O contracts, indicating that the single salts (Rb₂SO₄, Cs₂SO₄) are more readily precipitated at higher temperature, which provides theoretical guidance for the future production and separation of Rb, Cs, and Mg from sulfate-type salt lakes.

1. Introduction

Rubidium and cesium are essential minerals for advancing strategic emerging industries, including aerospace, defense, and information technology. They are also fundamental raw materials for modern high-tech research and development [1,2], which are crucial for advancing energy conversion and communication technologies. Rubidium and cesium, along with their compounds, are renowned for their exceptional photoelectric properties. These elements have a wide range of applications, spanning both traditional industries and emerging technologies. In traditional sectors, they are used in electronics, glass and ceramics, catalysts, biochemistry, and medicine. In addition, they are also finding extensive use in new fields such as energy conversion, communication technologies, and deep well drilling [3,4]. Despite China’s rich and varied resources of rubidium and cesium, the low grade of these elements in solid form poses significant challenges for their development and utilization. In contrast, rubidium and cesium are found in ionic form in salt lakes and geothermal waters in Xizang and Qinghai. Extracting these elements from such sources is relatively straightforward, with lower costs and energy consumption, making it a primary focus for future extraction efforts [5,6,7]. Challenges remain in the development and utilization of rubidium and cesium due to their low concentrations in salt lakes, the presence of high concentrations of interfering ions, and the propensity for these elements to be carried away or precipitated, which complicates the extraction process. In regions like Qinghai and Xizang, which are dominated by sulfate-type salt lakes, the concentration of Mg²⁺ is relatively high. The phase diagram of the water–salt system is instrumental in elucidating the solubility of different salts in water, the interactions between salts, and the crystallization behavior. It also helps in optimizing operational conditions and addressing crystallization or precipitation issues. This diagram provides a theoretical basis for studying new materials and technologies, guiding experimental and industrial applications [8]. Scholars have extensively studied rubidium and cesium. Potassium, rubidium, and cesium have similar ionic radii and tend to form solid solutions [9]: examples include (KCl)_x(RbCl)_1−x, (RbCl)_x(CsCl)_1−x, (KCl)_x(CsCl)_1−x [10,11,12], (K, Cs)₂SO₄, (Rb, Cs)₂SO₄, and (K, Rb)₂SO₄ [13,14,15,16]. Consequently, the phase diagram of the quaternary system Rb⁺, Cs⁺, Mg²⁺//SO₄²⁻ - H₂O was necessary to understand the influence of magnesium on the behavior of rubidium and cesium.

Rubidium, cesium, and other cations readily form double salts; the ternary system Rb⁺, Mg²⁺//SO₄²⁻ - H₂O results in a double salt Rb₂SO₄·MgSO₄·6H₂O; and in the Cs⁺, Mg²⁺//SO₄²⁻ - H₂O system, Cs₂SO₄·MgSO₄·6H₂O is present [9,17]. The ternary system Na⁺, Rb⁺//SO₄²⁻ - H₂O is classified as a hydrated I type at 298.2, 323.2, 348.2, and 363.2 K, producing the single salt Rb₂SO₄, while Na₂SO₄ transitions from Na₂SO₄·10H₂O(298.2 K) to Na₂SO₄(323.2, 348.2, 363.2 K). What is more, in the ternary system Na⁺, Cs⁺//SO₄²⁻ - H₂O, the system exhibits a hydrated II type at 298.2 K, forming the single salts Na₂SO₄, Na₂SO₄·10H₂O, and Cs₂SO₄. As the temperature increases (323.2, 348.2, 363.2 K), the system transitions to a hydrated I type, leading to the disappearance of the single salt Na₂SO₄·10H₂O [15,18]. The quaternary system K⁺, Rb⁺, Cs⁺//SO₄²⁻ - H₂O exhibited solid solutions of (K, Rb)₂SO₄, (K, Cs)₂SO₄, and (Rb, Cs)₂SO₄ [16]. As the temperature increased, the crystallization region of (K, Cs)₂SO₄ decreased while the crystallization regions of (K, Rb)₂SO₄ and (Rb, Cs)₂SO₄ increased. The different trends in the variation in the crystallization regions with temperature are attributed to the differences in the radii of the cations in the solid solutions. The quaternary system Rb⁺, Cs⁺, Mg²⁺//SO₄²⁻ - H₂O exhibits four invariant points and six crystalline phase regions at 298.2 K [19]. Previous studies have shown that rubidium (Rb) and cesium (Cs) readily form solid solutions or double salts with various ions. Additionally, the crystallization form of hydrated salts changes at different temperatures, which can affect the extraction and separation of rubidium and cesium. To clarify the precipitation behavior of these rare elements in sulfate-type salt lakes, the phase equilibria of the quaternary system Rb⁺, Cs⁺, Mg²⁺//SO₄²⁻ - H₂O at 323.2 K were investigated. Concurrently, the comparison of the system at 298.2 and 323.2 K was conducted.

2. Experimental Section

2.1. Reagents

Ultrapure water with a conductivity of κ ≤ 5.5 × 10⁻⁶ S·m⁻¹ was used in this experiment, and the detailed descriptions of the chemicals are provided in

Table 1. Phase equilibria experimental reagents.

Chemical Agents	CAS No.	Purity (w/w) a	Source	Analysis Method
Rb2SO4	7488-54-2	0.995	Jiangxi Dongpeng	alizarin red-S volumetric method [20]
Cs2SO4	10294-54-9	0.999	Jiangxi Dongpeng
MgSO4·7H2O	10034-99-8	0.999	Shanghai Bohr

^a The purity in mass fraction was stated by the suppliers.

.

2.2. Experimental Instruments

The experimental equipment used for the phase equilibrium studies is shown in

Table 2. Experimental instruments.

Experimental Instruments	Precision	Producer	Type Number	Usage
Analytical balance	0.0001 g	Sartorius	BSA124S	Weighing
Thermostat	±0.2 K	Chongqing Inborn Experiment Instrument	SHH250	Constant temperature
X-ray powder diffraction	1°	Dandong Fangyuan Instrument Co., Ltd.	DX-2700	Solid phase identification
Atomic absorption spectroscopy	0.0001	iCE 3300, ThermoFisher	iCE 3300	Liquid phase identification

.

2.3. Experimental Procedure and Analytical Method

The quaternary system Rb⁺, Cs⁺, Mg²⁺//SO₄²⁻ - H₂O includes the subsystems Rb⁺, Cs⁺//SO₄²⁻ - H₂O; Rb⁺, Mg²⁺//SO₄²⁻ - H₂O; and Cs⁺, Mg²⁺//SO₄²⁻ - H₂O. Based on the invariant point of the ternary subsystem, an initial mixed solution was prepared at 323.2 K. Then, a new salt with a mass fraction ranging from 0 to saturation was added, following the solubility of the new salt. All samples were placed in a thermostat maintained at 323.2 ± 0.2 K and oscillated for 45 days. Equilibrium was considered to be reached when the variance between adjacent sampling analyses of the same sample was less than 0.3%. Once equilibrium was achieved, oscillation was stopped, and the samples were allowed to settle, resulting in a transparent upper layer of solution and a lower layer of solid. The density of the supernatant was measured using the pycnometer method, and its composition was analyzed. The analytical methods employed for each ion in the liquid phase were as follows: sulfate [20]: alizarin red-S volumetric method, with a standard uncertainty of ±0.0037; Mg²⁺ [21]: EDTA volumetric method, with standard uncertainties of ±0.0049; Rb⁺, Cs⁺ [22]: Atomic absorption spectrophotometry (AAS), with standard uncertainties of ±0.0050 and ±0.0050. The standard uncertainties for the mass fraction of Rb₂SO₄, Cs₂SO₄, and MgSO₄ were ±0.0057, ±0.0057, and ±0.0052, respectively. Each determination was conducted three times to ensure the precision and reproducibility of the analytical data. The calculation formulas of the mass fraction w(B) and Jänecke index J(B) (mol/100 mol dry salt) (B = Rb₂SO₄, Cs₂SO₄, MgSO₄, H₂O) are as follows:

$${w(\mathrm{Rb}}_2{\mathrm{SO}}_4{)+w(\mathrm{Cs}}_2{\mathrm{SO}}_4{)+w(\mathrm{MgSO}}_4{)+w(\mathrm{H}}_2\mathrm{O})=100\%$$

(1)

$$\left[\mathrm{M}\right]={\displaystyle \frac{{w(\mathrm{Rb}}_2{\mathrm{SO}}_4)}{267.00}}+{\displaystyle \frac{{w(\mathrm{Cs}}_2{\mathrm{SO}}_4)}{361.88}}+{\displaystyle \frac{{w(\mathrm{MgSO}}_4)}{120.37}}$$

(2)

$${J(\mathrm{Rb}}_2{\mathrm{SO}}_4)={\displaystyle \frac{{w(\mathrm{Rb}}_2{\mathrm{SO}}_4)}{267.00 \times \left[\mathrm{M}\right]}} \times 100$$

(3)

$${J(\mathrm{Cs}}_2{\mathrm{SO}}_4)={\displaystyle \frac{{w(\mathrm{Cs}}_2{\mathrm{SO}}_4)}{361.88 \times \left[\mathrm{M}\right]}} \times 100$$

(4)

$${J(\mathrm{MgSO}}_4)={\displaystyle \frac{{w(\mathrm{MgSO}}_4)}{120.37 \times \left[\mathrm{M}\right]}} \times 100$$

(5)

$${J(\mathrm{H}}_2\mathrm{O})={\displaystyle \frac{{w(\mathrm{H}}_2\mathrm{O})}{18.02 \times \left[\mathrm{M}\right]}} \times 100$$

(6)

$${J(\mathrm{Rb}}_2{\mathrm{SO}}_4{)+J(\mathrm{Cs}}_2{\mathrm{SO}}_4{)+J(\mathrm{MgSO}}_4)=100$$

(7)

3. Results and Discussion

3.1. Phase Diagram of Quaternary System Rb⁺, Cs⁺, Mg²⁺//SO₄²⁻ - H₂O at 323.2 K

The quaternary system Rb⁺, Cs⁺, Mg²⁺//SO₄²⁻ - H₂O at 323.2 K is composed of three binary subsystems and three ternary subsystems: Rb⁺//SO₄²⁻ - H₂O; Cs⁺//SO₄²⁻ - H₂O; Mg²⁺//SO₄²⁻ - H₂O; Rb⁺, Cs⁺//SO₄²⁻ - H₂O; Cs⁺, Mg²⁺//SO₄²⁻ - H₂O; Rb⁺, Mg²⁺//SO₄²⁻ - H₂O. The invariant point data at 323.2 K from the literature are compared with the co-saturation point data obtained in this work, as listed in Figure 1 [14,16,23]. Figure 1 shows that the data for the above ternary subsystems obtained in this work are basically consistent with the literature data.

The phase equilibrium experimental data, including solubility, density, refractive index, and the composition of the equilibrium solid and liquid phases, are presented in

Table 3. The phase equilibrium experimental data, including solubility, density, refractive index, and the composition of the equilibrium solid and liquid phases of the quaternary system Rb⁺, Cs⁺, Mg²⁺//SO₄²⁻ - H₂O at 323.2 K.

No.	Density /ρ(g·cm−3)	Refractive Index	Equilibrium Solutions Composition, w(B) × 102				Jänecke Index of Dry Salt, J(B) (mol/100 mol S)				Equilibrium Solid Phase b
							J(Rb2SO4) + J(Cs2SO4) + J(MgSO4) = 100
			w(Rb2SO4)	w(Cs2SO4)	w(MgSO4)	w(H2O)	J(Rb2SO4)	J(Cs2SO4)	J(MgSO4)	J(H2O)
1,A	1.5309	1.3761	34.65	16.11	0.00	49.24	74.46	25.54	0.00	1568.20	Rb2SO4 + SS
2	1.5406	1.3800	34.38	16.12	0.65	48.85	72.05	24.93	3.02	1517.33	Rb2SO4 + SS
3,S1	1.5502	1.3804	34.46	15.70	1.36	48.48	70.24	23.61	6.15	1464.55	Rb2SO4 + SS + RM
4,B	2.0499	1.4109	1.51	65.40	0.00	33.09	3.03	96.97	0.00	985.51	Cs2SO4 + SS
5,S2	2.0794	1.4107	5.39	59.75	0.46	34.40	10.67	87.31	2.02	1009.68	Cs2SO4 + SS + CM
6,C	2.0338	1.4104	0.00	63.32	0.79	35.89	0.00	96.38	3.62	1097.40	Cs2SO4 + CM
7	2.0359	1.4110	3.04	60.08	0.69	36.19	6.22	90.65	3.13	1096.89	Cs2SO4 + CM
8	2.0594	1.4105	4.32	59.52	0.58	35.58	8.72	88.68	2.60	1064.84	Cs2SO4 + CM
9,S2	2.0794	1.4107	5.39	59.75	0.46	34.40	10.67	87.31	2.02	1009.68	Cs2SO4 + SS + CM
10,D	1.5375	1.4075	0.00	12.37	32.92	54.71	0.00	11.11	88.89	987.05	M6 + CM
11	1.5313	1.4073	2.89	8.97	33.02	55.12	3.49	8.00	88.51	987.20	M6 + CM
12,S3	1.5067	1.4074	3.92	7.61	33.06	55.41	4.73	6.78	88.49	991.01	M6 + RM + CM
13	1.4212	1.3913	6.87	7.32	24.43	61.38	10.34	8.13	81.54	1368.79	RM + CM
14	1.3397	1.3782	11.52	8.28	15.69	64.51	21.97	11.65	66.38	1823.49	RM + CM
15	1.3575	1.3690	17.50	9.46	8.82	64.22	39.73	15.85	44.42	2161.01	RM + CM
16	1.3779	1.3673	20.84	9.36	5.82	63.98	51.26	16.99	31.75	2332.36	RM + CM
17,S4	1.5169	1.3801	33.23	16.82	1.36	48.59	68.29	25.51	6.20	1480.05	SS + RM + CM
18,E	1.4616	1.4027	5.60	0.00	33.51	60.89	7.01	0.00	92.99	1129.03	M6 + RM
19	1.4866	1.4065	4.95	5.21	32.37	57.47	6.14	4.77	89.09	1056.82	M6 + RM
20,S3	1.5067	1.4074	3.92	7.61	33.06	55.41	4.73	6.78	88.49	991.01	M6 + RM + CM
21,F	1.4030	1.3704	38.06	0.00	1.43	60.51	92.31	0.00	7.69	2175.04	Rb2SO4 + RM
22	1.4687	1.3730	37.03	6.71	1.42	54.84	82.05	10.97	6.98	1800.94	Rb2SO4 + RM
23	1.5043	1.3825	36.48	10.37	1.39	51.76	77.26	16.21	6.53	1624.77	Rb2SO4 + RM
24	1.5228	1.3810	35.64	13.69	1.37	49.30	73.06	20.71	6.23	1497.90	Rb2SO4 + RM
25,S1	1.5502	1.3804	34.46	15.70	1.36	48.48	70.24	23.61	6.15	1464.55	Rb2SO4 + SS + RM
26,S4	1.5169	1.3801	33.23	16.82	1.36	48.59	68.29	25.51	6.20	1480.05	SS + RM + CM
27	1.5273	1.3812	32.43	17.45	1.31	48.81	67.27	26.71	6.03	1500.51	SS + CM
28	1.5931	1.3862	29.69	26.06	1.23	43.02	57.49	37.23	5.28	1234.55	SS + CM
29	1.7277	1.3910	24.80	33.67	1.14	40.39	47.54	47.62	4.85	1147.41	SS + CM
30	1.7820	1.3961	22.81	40.29	1.05	35.85	41.57	54.18	4.25	968.41	SS + CM
31	1.8435	1.4008	19.79	45.99	0.99	33.23	35.39	60.68	3.93	880.74	SS + CM
32	1.9604	1.4087	16.13	48.68	0.72	34.47	30.07	66.95	2.98	952.34	SS + CM
33	1.9833	1.4095	16.01	56.33	0.78	26.88	27.00	70.08	2.92	671.79	SS + CM
34	2.0278	1.4110	13.71	56.06	0.84	29.39	24.08	72.65	3.27	765.05	SS + CM
35	2.0586	1.4116	10.41	60.71	0.93	27.95	18.18	78.22	3.60	723.37	SS + CM
36,S2	2.0794	1.4107	5.39	59.75	0.46	34.40	10.67	87.31	2.02	1009.68	Cs2SO4 + SS + CM

^b Standard uncertainty u and relative standard uncertainty u_r: u(T) = 0.20 K; u(ρ) = 0.0002 g·cm⁻³; u(n_D) = 0.0002; u_r(Rb₂SO₄) = 0.0057; u_r(Cs₂SO₄) = 0.0057; u_r(MgSO₄) = 0.0052; M6: MgSO₄·6H₂O; SS: (Rb, Cs)₂SO₄; RM: Rb₂SO₄·MgSO₄·6H₂O; CM: Cs₂SO₄·MgSO₄·6H₂O.

.

Based on the experimental data presented in Table 3, the following figures were generated for the quaternary system Rb⁺, Cs⁺, Mg²⁺//SO₄²⁻ - H₂O at 323.2 K: the space stereogram (Figure 2), the phase diagram (Figure 3), the XRD pattern of the solid phases corresponding to the invariant point S₁–S₄ (Figure 4), the multi-temperature comparison diagram (Figure 5), the water content-composition diagram (Figure 6), the density-composition diagram (Figure 7), and the refractive index-composition diagram (Figure 8).

Figure 2 and Figure 3 reveal that the quaternary system Rb⁺, Cs⁺, Mg²⁺//SO₄²⁻ - H₂O at 323.2 K is a complex one, composed of a solid solution (Rb, Cs)₂SO₄, double salts (Cs₂SO₄·MgSO₄·6H₂O, Rb₂SO₄·MgSO₄·6H₂O), single salts (Rb₂SO₄, Cs₂SO₄), and a hydrated single salt (MgSO₄·6H₂O). The phase diagram includes four quaternary invariant points, nine univariate curves, and six crystallization regions.

The XRD patterns of the solid salt compositions corresponding to the invariant points S₁, S₂, S₃, and S₄ are depicted in Figure 4. The four invariant points are as follows: S₁: Rb₂SO₄ + (Rb, Cs)₂SO₄ + Rb₂SO₄·MgSO₄·6H₂O; S₂: Cs₂SO₄ + Cs₂SO₄·MgSO₄·6H₂O + (Rb, Cs)₂SO₄; S₃: MgSO₄·6H₂O + Rb₂SO₄·MgSO₄·6H₂O + Cs₂SO₄·MgSO₄·6H₂O; S₄: (Rb, Cs)₂SO₄ + Cs₂SO₄·MgSO₄·6H₂O + Rb₂SO₄·MgSO₄·6H₂O.

The nine univariate curves (AS₁, BS₂, CS₂, DS₃, ES₃, FS₁, S₁S₄, S₂S₄, and S₃S₄) were co-saturated with two salts and an equilibrated solution, and the saturated salts for each isothermal dissolution curve are mentioned as follows: (1) AS₁: Rb₂SO₄ + (Rb, Cs)₂SO₄; (2) BS₂: Cs₂SO₄ + (Rb, Cs)₂SO₄; (3) CS₂: Cs₂SO₄ + Cs₂SO₄·MgSO₄·6H₂O; (4) DS₃: MgSO₄·6H₂O + Cs₂SO₄·MgSO₄·6H₂O; (5) ES₃: MgSO₄·6H₂O + Rb₂SO₄·MgSO₄·6H₂O; (6) FS₁: Rb₂SO₄ + Rb₂SO₄·MgSO₄·6H₂O; (7) S₁S₄: (Rb, Cs)₂SO₄ + Rb₂SO₄·MgSO₄·6H₂O; (8) S₂S₄: (Rb, Cs)₂SO₄ + Cs₂SO₄·MgSO₄·6H₂O; (9) S₃S₄: Rb₂SO₄·MgSO₄·6H₂O + Cs₂SO₄·MgSO₄·6H₂O. The six crystallization regions decrease in the order of Cs₂SO₄·MgSO₄·6H₂O, Rb₂SO₄·MgSO₄·6H₂O, (Rb, Cs)₂SO₄, Rb₂SO₄, Cs₂SO₄, and MgSO₄·6H₂O, and the solubility of the corresponding salt increases in this order.

The solubility of salts in water is affected by two sets of factors. Internal factors include ion structure, crystal form, radius, charge number, etc., while external factors include temperature, pH, pressure, etc. Rubidium and cesium, with their similar ionic radius and unit cell parameters, are prone to isomorphous substitution. This study found that the single salts Rb₂SO₄ and Cs₂SO₄ form the solid solution (Rb, Cs)₂SO₄ at 323.2 K. Compared with the other temperatures, it shows that the solid solution can exist stably in the research temperature range (298.2K~323.2K). According to the judgment rules of commensurate-type and incommensurate-type invariant points, it can be observed that the invariant points S₁, S₂, and S₃ belong to the commensurate-type invariant points, while the invariant point S₄ belongs to the incommensurate-type invariant point [24].

3.2. Comparison of Rb⁺, Cs⁺, Mg²⁺//SO₄²⁻ - H₂O at 298.2 K and 323.2 K

Based on the experimental data presented in Table 3 and existing research [19], multi-temperature comparison diagrams for the quaternary system Rb⁺, Cs⁺, Mg²⁺//SO₄²⁻ - H₂O at 298.2 K and 323.2 K have been created and are displayed in Figure 5. The figure indicates that the quaternary system comprises two hydrated double salts (Cs₂SO₄·MgSO₄·6H₂O and Rb₂SO₄·MgSO₄·6H₂O) and one solid solution [(Rb, Cs)₂SO₄]. It is classified as a complex quaternary system, with its phase diagram containing four invariant points, nine univariate curves, and six crystallization regions. Research has demonstrated that the crystalline form of magnesium sulfate is susceptible to temperature changes. It exists in the form of MgSO₄·7H₂O at 298.2 K and MgSO₄·6H₂O at 323.2 K. As the temperature increases, the crystalline phases of the single salts Cs₂SO₄ and Rb₂SO₄, the solid solution (Rb, Cs)₂SO₄, and the double salt Cs₂SO₄·MgSO₄·6H₂O also increase. This suggests that heating is advantageous for the separation of valuable single salts.

3.3. Physicochemical Properties of Quaternary System Rb⁺, Cs⁺, Mg²⁺//SO₄²⁻ - H₂O at 323.2 K

Referencing Table 3 and Figure 6, the water content diagram reveals that, on the univariate curve BS₂(CS₂), the water content decreases as the J(Rb₂SO₄) value increases. This trend is attributed to the addition of MgSO₄(Rb₂SO₄), which raises the salt content in the system. Conversely, on the univariate curves DS₃ and S₃E, the water content increases with the rise of J(Rb₂SO₄). Similarly, on the univariate curves S₁A and S₁F, the water content also increases with the increase in J(Rb₂SO₄). On the univariate curve S₂S₄, the water content first increases and then decreases with the increasing J(Rb₂SO₄), whereas on the univariate curve S₃S₄, the water content first decreases and then increases with the increasing J(Rb₂SO₄).

As depicted in the density-composition diagram (Figure 7), the density decreases with the increase in J(Rb₂SO₄) along the univariate curves DS₃, S₃E, S₁A, S₁F, and S₂S₄. Conversely, on the univariate curves CS₂, BS₂, and S₄S₁, the density increases with the increase in J(Rb₂SO₄). On the univariate curve S₃S₄, the density behaves differently; it decreases initially with the increase in J(Rb₂SO₄) and then increases, corresponding to the pattern where the water content in the solution first increases and then decreases.

The refractive index-composition diagram (Figure 8) indicates that, on the univariate curves BS₂, S₂S₄, S₃E, S₁F, and DS₃, the refractive index decreases as the J(Rb₂SO₄) value increases. On the univariate curves CS₂, S₄S₁, and S₁A, the refractive index decreases with the increase in J(Rb₂SO₄). On the univariate curve S₃S₄, the refractive index is influenced by the variation in water content, showing a decrease followed by an increase in J(Rb₂SO₄). Overall, the general trend of the refractive index is largely similar to that of the density.

4. Conclusions

The phase equilibria of the quaternary system Rb⁺, Cs⁺, Mg²⁺//SO₄²⁻ - H₂O at 323.2 K were investigated using the isothermal dissolution equilibrium method, and the phase diagrams of the system at different temperatures were compared. The conclusions are as follows. The quaternary system Rb⁺, Cs⁺, Mg²⁺//SO₄²⁻ - H₂O at 323.2 K is characterized by four invariant points, nine univariant curves, and six crystalline regions. The crystalline phases identified are Rb₂SO₄, Cs₂SO₄, MgSO₄·6H₂O, (Rb, Cs) ₂SO₄, Rb₂SO₄·MgSO₄·6H₂O, and Cs₂SO₄·MgSO₄·6H₂O, with the largest area corresponding to Cs₂SO₄ MgSO₄·6H₂O. At 298.2 and 323.2 K, the crystal form of MgSO₄ changed from MgSO₄·7H₂O (298.2 K) to MgSO₄·6H₂O (323.2 K). With the increase in temperature, the crystal phase region of Rb₂SO₄, Cs₂SO₄, and Cs₂SO₄·MgSO₄·6H₂O increased, and the crystal phase region of Rb₂SO₄·MgSO₄·6H₂O decreased. This indicates that, as the temperature rises, Rb and Cs are more likely to increase in the form of a single salt (Rb₂SO₄, Cs₂SO₄), enhancing the separation efficiency of these rare and valuable elements. The phase diagram reveals the interaction relationships between rubidium, cesium, and magnesium. It can serve as a guide for different concentrations and temperatures during mining, clarifying the changes in precipitated salts at each stage, and providing theoretical guidance for subsequent high-value utilization.