Phase Equilibria in the Quaternary Systems NaOH + Na₂CO₃ + Na₂SO₄ + H₂O, Na₂CO₃ + NaOH + NaCl + H₂O, and NaOH + Na₂SO₄ + NaCl + H₂O at 363.15 K

Abstract

The solid–liquid equilibrium data of the aqueous NaOH–Na₂CO₃–Na₂SO₄–H₂O, NaOH–Na₂CO₃–NaCl–H₂O, and NaOH–Na₂SO₄–NaCl–H₂O quaternary systems at 363.15 K were measured. The equilibrium solid phases and solubilities of salts in the three systems and its subsystems were determined. The densities of the saturated solutions were also determined. The experimental data are used to plot the solubility diagrams and water content diagrams of the systems. It was found that the NaOH–Na₂CO₃–Na₂SO₄–H₂O system contains the solid solution of γ-salt (mNa₂SO₄·nNa₂CO₃) and the other two systems Na₂CO₃–NaOH–NaCl–H₂O and NaOH–Na₂SO₄–NaCl–H₂O have the complex salts S1 (Na₂SO₄·NaOH) and S3 (Na₂SO₄·NaCl·NaOH). On the basis of Xu’s activity coefficient model, a model was constructed for the correlation of solid–liquid equilibrium in electrolyte solutions to calculate the solubilities of salts in these systems at 363.15 K. The calculated solubilities are in agreement with the experimental values.

1. Introduction

There are abundant resources of natural soda in Inner Mongolia and Henan Province in China. The major components of natural soda include sodium carbonate (Na₂CO₃), sodium bicarbonate (NaHCO₃), sodium sulfate (Na₂SO₄), and sodium chloride (NaCl).² Based on these components, natural soda can be used to produce caustic soda. After natural soda is dissolved in water, insoluble impurities can be removed through the flocculation clarification process. When lime milk is added, which reacts with sodium carbonate to form a caustic solution, the NaOH concentration in caustic liquor is usually approximately 8–10%. Considering the economic efficiency and equilibrium conversion rate (the causticization rate is generally controlled at approximately 90%), as well as the fact that only a very small amount of sodium bicarbonate (NaHCO₃) is retained, diluted caustic soda liquid can be regarded as a quinary system of NaOH–NaCl–Na₂CO₃–Na₂SO₄–H₂O.

To obtain caustic soda products that meet quality standards, the caustic soda liquid should be concentrated, and Na₂SO₄, NaCl, and a small amount of Na₂CO₃ impurities should be removed. The evaporation method for producing caustic soda through the causticizing process generally involves three-effect evaporation. The first-effect temperature is approximately 423.15 K, at which the concentration of caustic soda can be controlled at 11–12% and the diluted caustic soda liquid is concentrated; the second-effect temperature is approximately 393.15 K, at which the concentration of caustic soda is generally controlled at approximately 19–20% to achieve partial removal of the impurities; and the third-effect temperature is approximately 363.15 K, at which the concentration of caustic soda is controlled at 45–48%. At the third-effect temperature, Na₂SO₄ and NaCl³ are removed to concentrate the diluted caustic soda liquid and remove impurities; then, the caustic soda liquid is allowed to cool to achieve separation, dehydration, and solidification to form solid caustic soda that meets the quality requirements. In the actual production process, only the third-effect temperature is controlled at 363.15 K; at this point, the evaporated liquor has reached saturation and salting-out. Therefore, 363.15 K was chosen as the research temperature in this paper.

Liu et al.⁴ systematically studied the phase diagram of the quinary system NaOH–NaCl–Na₂CO₃–Na₂SO₄–H₂O at 423.15 K. Stephen et al.^5,6 collected detailed phase equilibrium data for the binary and ternary subsystems related to the quinary system under consideration at 363.15 K. Su et al.^7,8 studied a related quaternary system and obtained partial solubility data. To date, the phase equilibrium data for the quaternary subsystems of the quinary system NaOH–NaCl–Na₂CO₃–Na₂SO₄–H₂O, including NaOH–Na₂CO₃–Na₂SO₄–H₂O, NaOH–NaCl–Na₂CO₃–H₂O, NaOH–NaCl–Na₂SO₄–H₂O, and NaCl–Na₂CO₃–Na₂SO₄, are yet to be perfected. Xu et al.¹ proposed a new hypothesis for the reference state of activity coefficients; literature data for single electrolyte solutions and mixed electrolyte solution systems, NaCl–H₂O, Na₂SO₄–H₂O, and NaCl–Na₂SO₄–H₂O, with temperatures spanning from 273.15 to 373.15 K, were successfully correlated using Xu’s activity coefficient model.

The purpose of the present work is to determine the phase equilibrium data and the characteristics of the ternary and quaternary subsystems of the NaOH–NaCl–Na₂CO₃–Na₂SO₄–H₂O quinary system at 363.15 K to clarify the appropriate concentration of NaOH in the evaporated solution in caustic solution evaporation and provide basic data for the production of caustic soda through causticization. Moreover, the solubilities of these subsystems at 363.15 K are calculated using Xu’s activity coefficient model.

2. Experimental Section

2.1. Reagents and Instruments

Distilled water with a conductivity of less than 1.3 × 10^–4 S·m^–1 and a pH of 6 was used for solubility measurements and chemical analysis. All chemicals used were of analytical purity grade and recrystallized before use. The sources, purity, and CAS numbers are listed in Table 1.

Table 1. Purity and Suppliers of Chemicals.

chemicals name	molecular formula	mass fraction puritya (%)	recrystallization	CAS no.	source
sodium chloride	NaCl	≥99.5	dried in an oven at 723.15 K	7647-14-5	Tianjin Yongsheng Chemical Reagent Co., Ltd, China
sodium hydroxide	NaOH	≥96.0		1310-73-2	Tianjin Yongsheng Chemical Reagent Co., Ltd, China
sodium carbonate	Na2CO3	≥99.8	dried in an oven at 543.15 K	497-19-8	Tianjin Yongsheng Chemical Reagent Co., Ltd, China
sodium sulfate	Na2SO4	≥99.0	dried in an oven at 423.15 K	7757-82-6	Tianjin Yongsheng Chemical Reagent Co., Ltd, China
potassium chromate	K2CrO4	≥99.5		7789-00-6	Tianjin Yongsheng Chemical Reagent Co., Ltd, China
silver nitrate	AgNO3	≥99.8		7761-88-8	Sinopharm Chemical Regent Co., Ltd
phenolphthalein indicator	C20H14O4	≥98.0		77-09-8	Tianjin Yongsheng Chemical Reagent Co., Ltd, China
methyl orange indicator	C14H14N3NaO3S	≥96.0		547-58-0	Tianjin Chemical Reagent 3rd Factory
methyl red indicator	C15H15N3O2	AR		493-52-7	Tianjin Tianxin Chemical Development Center
barium chloride	BaCl2	≥99.5		10361-37-2	Tianjin Chemical Reagent 1st Factory
hydrochloric acid	HCl	≥38.0		7647-01-0	Tianjin Chemical Reagent 3rd Factory

^aStated by the suppliers.

A thermostatic water bath (76-1, Jiangsu Tianyou Co., Ltd.) with a precision of ±0.1 K was used for the solid–liquid equilibrium (SLE) measurements. In addition, we used a thermostatic water bath (KSA-II thyristor dc governor, 2.2 kW, Ningbo Beilun District Zhitou Electronic Control Equipment Factory), circulating heated water, to achieve a uniform mixture of the solid and liquid phases. An analytical balance, with an accuracy of ±0.001 g (ALC-110.4, Sartorius AG, Germany), was used for weighing; a muffle furnace (SX-1300 °C, Tianjin Zhonghuan Experimental Electric Furnace Co., Ltd.) was used to burn the precipitate; and an intelligent temperature controller (XMTG-6000, Jiangsu Taizhou Haige Instrument Co., Ltd.), which was calibrated against a temperature calibrator, was used.

2.2. Experimental Methods

The system points were prepared as follows. For the ternary systems, the system points were prepared by gradually adding the second component on the basis of the single salt saturation points. For the quaternary systems, the system points were prepared by adding the third component on the basis of the double salt cosaturation points. Each solubility curve was prepared using 5–15 groups, and an appropriate amount of distilled water was added to each group of mixtures. The matched experimental solutions were put into glass bottles (30 mL, 3.5 cm in diameter, and 6.5 cm high), sealed with rubber plugs and aluminum caps, placed in a thermostatic water bath (363.15 ± 0.1 K) with a rotating drum, and rotated at a constant speed in the thermostatic water bath to achieve phase equilibrium.⁹

The experimental setup, which was established in accordance with the characteristics of the studied system, is shown in Figure 1. The balancing device could also be used for static and sampling operations to prevent temperature changes in the external environment from disturbing the balance of the samples. A micro ac constant-speed motor made of plexiglass was used at a speed of 60 rpm to achieve liquid–solid phase equilibrium. The equilibrium time was approximately three days; after equilibrium was achieved, the supernatant liquid was removed from the balanced bottle every few hours and chemical analysis was conducted. If the relative error among the three consecutive sampling was below 0.003, the equilibrium could be considered to be achieved. It was determined using the experimental results with the equilibrium time and static time in Table 2. At equilibrium, the sample solutions were allowed to rest for 3 days to separate the solid and liquid phases and kept in a constant-temperature oil bath. The upper portion of the sealed bottles was kept above the liquid level to prevent oil from entering the balanced bottles during sampling and then affecting the experimental results. After reaching equilibrium, the sampling tubes were preheated in an oven to avoid disturbing the equilibrium, and then the supernatant was used to determine the chemical composition of the equilibrium liquid phase via chemical analysis.¹⁰

Figure 1.

Schematic diagram of the experimental setup for phase equilibrium determinations. (1) Bearing support; (2) belt pulley; (3) rotor drum; (4) thermoelectric couple; (5) dc motor; (6) thermometer; and (7) thermostatic bath.

Table 2. Determination of the Experimental Results with the Equilibrium Time and Static Time.

	fluid mass, w(B) × 100a, %
no.	NaOH	Na2CO3	Na2SO4	NaCl	H2O	equilibrium phase solidsb	equilibrium time	static time
1	34.18	0.00	2.22	0.00	63.60	S + S1	24	1
	34.16	0.00	2.16	0.00	63.68		22	1.5
	34.21	0.00	2.30	0.00	63.49		22	2
2	51.00	2.05	0.00	4.01	42.94	C + OH + Cl	72	72
	51.04	2.05	0.00	3.99	42.92		74	72
	50.97	2.11	0.00	4.03	42.89		76	72

^aw(B) is the mass fraction of the component (B).

^bAbbreviations: Cl: NaCl; S: Na₂SO₄; OH: NaOH; S1: NaOH·Na₂SO₄; and C: Na₂CO₃.

To prevent effects from CO₂ in the air, the samples were collected in a nitrogen cabinet. The contents of CO₂ in the vapor phase were quite small in most cases: the ratio of the partial pressure to the total pressure in the equilibrium vapor phase was only 0.09–0.25%. In addition, the experimental systems were sealed and the volumes of the vapor phases were much smaller than those of the liquid and solid phases. Accordingly, the formation of CO₂ had almost no effect on the equilibrium fluid composition.

2.3. Analytical Methods

2.3.1. Liquid-Phase Analytical Methods

According to a previously reported method,^11,12 the total alkali (sodium carbonate and sodium hydroxide) content was titrated with standard hydrochloric acid using methyl orange solution as an indicator. The content of NaOH was determined by excess alkalimetry using phenolphthalein solution as an indicator. The chlorine ion concentration was measured by silver nitrate titration. The content of sulfate ions was determined by a gravimetric method. The details of the abovementioned analytical methods can be found in the literature.¹³ The solution density was measured by the density bottle method with a precision of ±0.0002 g. Each experiment was performed three times in parallel, and a relative standard uncertainty of 0.05 was achieved.^14−17,21 The results of the repeatability experiments shown in Table 3 confirmed that the phase equilibrium data measured by the setup in this work had good repeatability.

Table 3. Repeatability of Data of Phase Equilibria Measured at T = 363.15 K and P = 88.94 kPa^a.

		liquid mass, w(B) × 100b, %
cosaturated points	serial number	Na2CO3	NaCl	Na2SO4	NaOH	H2O	equilibrium phase solidsc
Table 5 9(E1)	1	0	0	2.21	34.18	63.61	S + S1
	2	0	0	2.30	34.21	63.49
	3	0	0	2.16	34.16	63.68
	average value	0	0	2.22	34.18	63.60
15(E2)	1	0	0	2.71	71.62	25.67	S1 + OH
	2	0	0	2.74	71.59	25.67
	3	0	0	2.79	71.64	25.57
	average value	0	0	2.75	71.62	25.63
Table 6 12(E)	1	0	2.85	0	68.89	28.26	Cl + OH
	2	0	2.89	0	68.92	28.19
	3	0	2.87	0	68.81	28.32
	average value	0	2.87	0	68.87	28.26
Table 7 14(E1)	1	2.22	0.00	6.54	26.98	64.26	S + γ + S1
	2	2.34	0.00	6.59	27.07	64.00
	3	2.31	0.00	6.70	27.04	63.95
	average value	2.29	0.00	6.61	27.03	64.07
15(E2)	1	4.89	0.00	1.83	61.82	31.46	OH + C1 + γ
	2	4.79	0.00	1.77	61.83	31.61
	3	4.98	0.00	1.75	61.89	31.38
	average value	4.89	0.00	1.78	61.85	31.48
16(E3)	1	2.89	0.00	0.83	51.66	44.62	C1 + C + OH
	2	2.88	0.00	0.86	51.69	44.57
	3	2.81	0.00	0.82	51.64	44.73
	average value	2.86	0.00	0.84	51.66	44.64
17(E4)	1	2.05	0.00	1.63	55.79	40.53	OH + S1 + γ
	2	2.12	0.00	1.68	55.76	40.44
	3	2.10	0.00	1.75	56.01	40.14
	average value	2.09	0.00	1.69	55.85	40.37
Table 8 5(E1)	1	2.05	3.99	0.00	51.04	42.92	C + OH + Cl
	2	2.11	4.03	0.00	50.97	42.89
	3	1.99	4.00	0.00	51.00	43.01
	average value	2.05	4.01	0.00	51.00	42.94
7(E2)	1	1.66	4.35	0.00	34.18	59.81	C + C1 + Cl
	2	1.71	4.37	0.00	34.22	59.70
	3	1.69	4.29	0.00	34.23	59.79
	average value	1.69	4.34	0.00	34.21	59.76
Table 9 4(E1)	1	0.00	0.71	3.12	41.85	54.32	S + S1 + S3
	2	0.00	0.69	3.12	41.79	54.40
	3	0.00	0.73	3.15	41.92	54.20
	average value	0.00	0.71	3.13	41.85	54.31
6(E2)	1	0.00	4.51	3.50	60.36	31.63	S + Cl + S3
	2	0.00	4.66	3.43	60.40	31.51
	3	0.00	4.62	3.34	60.55	31.49
	average value	0.00	4.60	3.42	60.44	31.54
9(E3)	1	0.00	2.38	2.58	81.31	13.73	Cl + S3 + OH
	2	0.00	2.37	2.51	81.29	13.83
	3	0.00	2.25	2.64	81.25	13.86
	average value	0.00	2.33	2.58	81.28	13.81
11(E4)	1	0.00	1.07	3.71	78.12	17.10	S1 + S3 + OH
	2	0.00	1.07	3.68	78.11	17.14
	3	0.00	1.06	3.72	78.13	17.09
	average value	0.00	1.07	3.70	78.12	17.11

^aStandard uncertainties u are u(T) = 0.1 K, u(P) = 0.5 kPa, and u_r(Na₂CO₃) = u_r(Na₂SO₄) = u_r(NaCl) = u_r(NaOH) = 0.05.

^bw(B) is the mass fraction of the component (B).

^cAbbreviations: S, Na₂SO₄; OH, NaOH; Cl, NaCl; C₁, Na₂CO₃·H₂O; S1, Na₂SO₄·NaOH; γ, mNa₂SO₄·nNa₂CO₃; and S3, Na₂SO₄·NaCl·NaOH.

2.3.2. Identification of the Solid Phase

The solids were analyzed by the wet residue method. The wet residue point and the liquid phase point were on the same line, and the equilibrium solid point was on the extension line. The solids were then analyzed by the abovementioned method to obtain the composition of the solid phases. In addition, the crystalloid form could be identified by an auxiliary method; that is, the wet solid sample was further prepared by drying with filter paper and studied by X-ray powder diffraction (XRD).¹⁸ To confirm the reliability of the experimental apparatus and method utilized in this article, before the formal experiment, we measured the data for one cosaturation point of the system NaCl–NaOH–H₂O with the experimental apparatus developed. The results (shown in Table 4) were in good accordance with the data in the literature.^5,6

Table 4. Comparison of Data of Phase Equilibria Measured and Literature Data at T = 363.15 K and P = 88.94 kPa^a.

		fluid mass, w(B) × 100, %
subject		Na2CO3	NaCl	Na2SO4	NaOH	solidc	data sources
Table 6 12(E)	measured results	0	2.87	0	68.87	Cl + OH	this work
	literature datab	0	2.90	0	65.00	Cl + OH	b
	relative error (%)d	0	0.01	0	0.06
	absolute errord	0	0.03	0	3.87

^aStandard uncertainties u are u(T) = 0.1 K, u(P) = 0.5 kPa, and u_r(Na₂CO₃) = u_r(Na₂SO₄) = u_r(NaCl) = u_r(NaOH) = 0.05.

^bData source: Stephen et al. (1963).

^cAbbreviations: OH, NaOH and Cl, NaCl.

^dε_r(%) = |measured-lit.|/lit.; ε_a = |measrd-lit.|.

3. Results and Discussion

3.1. Solid–Liquid Phase Equilibrium

3.1.1. Ternary System NaOH–Na₂SO₄–H₂O at 363.15 K

The measured solubilities of the ternary system NaOH–Na₂SO₄–H₂O at 363.15 K are listed in Table 5. The solubility of the equilibrated liquid phase is expressed as the mass fraction, and the corresponding phase diagram is plotted in Figure 2. There are two cosaturation points, E₁ and E₂, and two solubility curves, DE₁ and E₁E₂. Figure 3 shows the XRD patterns of the cosaturation points in this system. According to this figure, points E₁ and E₂ correspond to the cosaturation points of Na₂SO₄ + NaOH·Na₂SO₄ and NaOH·Na₂SO₄ + NaOH, respectively.

Table 5. Equilibrium Solubilities of the Ternary System NaOH–Na₂SO₄–H₂O at T = 363.15 K and P = 88.94 kPa^a.

	composition of liquid phase, w(B) × 100b, %			composition of wet residue, w(B) × 100b, %
no.	NaOH	Na2SO4	H2O	NaOH	Na2SO4	H2O	solution density ρ/g·cm–3	equilibrium phase solidsc
1	0.00	29.91	70.09	0.00	32.01	67.99	1.1984	S
2	6.47	22.86	70.67	4.69	46.87	48.44	1.2158	S
3	10.95	16.25	72.80	4.67	65.08	30.25	1.2034	S
4	14.79	12.41	72.80	7.99	52.67	39.34	1.1965	S
5	17.12	9.65	73.23	9.99	46.34	43.67	1.1990	S
6	20.84	6.07	73.09	14.52	35.32	50.16	1.2146	S
7	25.44	4.97	69.59	16.99	36.12	46.89	1.2522	S
8	30.24	2.49	67.27	22.30	27.86	49.84	1.2930	S
9(E1)	34.18	2.22	63.60	22.61	35.85	41.54	1.3321	S + S1
10	37.39	2.49	60.12	31.93	17.68	50.39	1.3657	S1
11	37.74	2.31	59.95	37.51	29.70	32.79	1.3686	S1
12	41.63	1.66	56.71	40.63	28.33	31.04	1.4048	S1
13	43.74	2.22	54.04	41.87	22.41	35.72	1.4279	S1
14	63.39	2.49	34.12	54.67	23.27	22.06	1.5884	S1
15(E2)	71.62	2.75	25.63	68.31	8.14	23.55	1.5884	S1 + OH
16	76.28	0.00	23.72	72.26	6.81	20.93	1.5884	OH

^aStandard uncertainties u are u(T) = 0.1 K, u(P) = 0.5 kPa, and u_r(Na₂SO₄) = u_r(NaOH) = 0.05.

^bw(B) is the mass fraction of the component (B).

^cAbbreviations: S, Na₂SO₄; OH, NaOH; and S1, Na₂SO₄·NaOH.

Figure 2.

Equilibrium phase diagrams of the ternary system NaOH–Na₂SO₄–H₂O at 363.15 K and 88.94 kPa; (■) measured solubility; (—) solubility curve.

Figure 3.

X-ray diffraction photograph of the cosaturation points E₁ and E₂ of the ternary system NaOH–Na₂SO₄–H₂O at 363.15 K.

3.1.2. Ternary System NaOH–NaCl–H₂O at 363.15 K

The measured solubilities of the ternary system NaCl–NaOH–H₂O at 363.15 K are presented in Table 6. The compositions of the saturated solution and wet solid phase are expressed as mass fractions. The corresponding phase diagram is shown in Figure 4. There is one cosaturation point, E, and two crystallization regions, CDE and EBF. Point E corresponds to the cosaturation of NaCl + NaOH. The two solubility curves of this ternary system are represented by curves DE and FE. The crystallization field of NaCl (EBFE) is larger than that of NaOH (CDEC). The XRD pattern of the cosaturation point (point E) is given in Figure 5.

Table 6. Equilibrium Solubilities of the Ternary System NaCl–NaOH–H₂O at T = 363.15 K and P = 88.94 kPa^a.

	composition of liquid phase, w(B) × 100b, %			composition of wet residue, w(B) × 100b, %
no.	NaCl	NaOH	H2O	NaCl	NaOH	H2O	solution density ρ/g·cm–3	equilibrium phase solidsc
1	27.80	0.00	72.2	46.34	0.00	53.66	1.1723	Cl
2	24.09	4.65	71.26	63.49	2.31	34.20	1.1935	Cl
3	20.28	9.95	69.77	67.11	3.67	29.22	1.2195	Cl
4	16.65	14.72	68.63	47.89	9.18	42.93	1.2445	Cl
5	11.96	24.28	63.76	45.78	14.80	39.42	1.3170	Cl
6	7.16	34.34	58.50	40.49	21.58	37.93	1.3708	Cl
7	5.30	39.91	54.79	39.67	25.17	35.16	1.4139	Cl
8	3.67	49.18	47.15	56.67	22.15	21.18	1.5019	Cl
9	2.63	55.81	41.56	57.78	23.77	18.45	1.5444	Cl
10	2.13	60.04	37.83	54.38	28.22	17.40	1.5763	Cl
11	2.09	65.61	32.30	68.99	21.85	9.16	1.5831	Cl
12(E)	2.87	68.87	28.26	41.55	41.42	17.03	1.5831	Cl + OH
13	2.30	70.17	27.53	2.24	74.64	23.12	1.5831	OH
14	2.32	71.56	26.12	1.74	75.65	22.61	1.5831	OH
15	1.60	73.52	24.88	1.48	75.20	23.32	1.5831	OH
16	0.00	76.28	23.72	0.00	76.42	23.58	1.5884	OH

^aStandard uncertainties u are u(T) = 0.1 K, u(P) = 0.5 kPa, and u_r(NaCl) = u_r(NaOH) = 0.05.

^bw(B) is the mass fraction of the component (B).

^cAbbreviations: OH, NaOH and Cl, NaCl.

Figure 4.

Equilibrium phase diagrams of the ternary system NaCl–NaOH–H₂O at 363.15 K and 88.94 kPa; (■) measured solubility; (—) solubility curve.

Figure 5.

X-ray diffraction photograph of the cosaturation point E of the ternary system NaCl–NaOH–H₂O at 363.15 K.

3.1.3. NaOH–Na₂CO₃–Na₂SO₄–H₂O System at 363.15 K

The experimental solubilities and equilibrium solids of the quaternary system NaOH–Na₂CO₃–Na₂SO₄–H₂O at 363.15 K are listed in Table 7. Based on the Jänecke dry salt indices, the dry salt solubility and water diagrams are plotted in Figure 6. A solid solution (γ, mNa₂SO₄·nNa₂CO₃) without complex salt was formed. Points E₁, E₂, E₃, and E₄ are the cosaturation points representing the saturation of Na₂SO₄ + γ + Na₂SO₄·NaOH, NaOH + Na₂SO₄·NaOH + γ, NaOH + Na₂CO₃ + Na₂CO₃·H₂O, and NaOH + Na₂CO₃·H₂O + γ, respectively. The system also contains six crystallization fields, which are represented by NaOH, Na₂CO₃, Na₂CO₃·H₂O, Na₂SO₄, S1(Na₂SO₄·NaOH), and γ-salt. The γ-salt has the largest crystallization field, indicating that it has the lowest solubility and thus can be readily crystallized out. The XRD patterns of points E₁, E₂, E₃, and E₄ are given in Figure 7.

Table 7. Experimental Values of Solubility and Dry Salt of the Quaternary System NaOH–Na₂CO₃–Na₂SO₄–H₂O at T = 363.15 K and P = 88.94 kPa^a.

	liquid mass, w(B) × 100b, %				dry salt (liquid) mass, w(B) × 100b, %
no.	NaOH	Na2CO3	Na2SO4	H2O	NaOH	Na2CO3	Na2SO4	H2O mass (g/100 g dry salt)	solution density ρ/g·cm–3	equilibrium phase solidsb
1	0.00	0.00	29.91	70.09	0.00	0.00	100.00	234.34	1.1984	S
2	34.18	2.22	0.00	63.60	93.90	6.10	0.00	174.73	1.3321	S1 + S
3	71.62	2.75	0.00	25.63	96.30	3.70	0.00	34.46	1.5885	S1 + OH
4	76.28	0.00	0.00	23.72	100.00	0.00	0.00	31.10	1.5884	OH
5	48.79	0.55	0.00	50.66	98.88	1.12	0.00	102.68	1.4696	C + OH
6	17.14	10.99	0.00	71.87	60.93	39.07	0.00	255.49	1.2339	C + C1
7	0.00	30.50	0.00	69.50	0.00	100.00	0.00	227.87	1.3112	C1
8	0.00	20.65	6.40	72.95	0.00	76.34	23.66	269.69	1.2557	C1 + γ
9	0.00	4.41	25.99	69.60	0.00	14.51	85.49	228.95	1.2248	S + γ
10	3.85	3.60	23.08	69.47	12.61	11.79	75.60	227.49	1.1948	S + γ
11	8.81	2.96	19.79	68.44	27.92	9.37	62.71	216.80	1.2193	S + γ
12	12.07	3.54	17.23	67.16	36.93	10.78	52.29	203.66	1.2186	S + γ
13	21.89	2.29	10.22	65.60	63.64	6.66	29.70	190.66	1.2574	S + γ
14(E1)	27.03	2.29	6.61	64.07	75.23	6.37	18.40	178.38	1.2869	S + γ + S1
15(E2)	55.85	2.09	1.69	40.37	93.66	3.51	2.83	67.71	1.5477	OH + S1 + γ
16(E3)	51.66	2.86	0.84	44.64	93.32	5.16	1.52	80.66	1.5102	C1 + C + OH
17(E4)	61.85	4.89	1.78	31.48	90.27	7.14	2.59	45.94	1.5885	OH + C1 + γ
18	21.55	5.08	2.25	71.12	74.62	17.58	7.80	246.24	1.2525	C1 + γ
19	11.80	13.16	4.28	70.76	40.36	45.00	14.64	242.08	1.2192	C1 + γ
20	10.64	13.86	4.54	70.96	36.65	47.72	15.63	244.39	1.2203	C1 + γ
21	8.64	15.02	5.05	71.29	30.10	52.32	17.58	248.29	1.2216	C1 + γ

^aStandard uncertainties u are u(T) = 0.1 K, u(P) = 0.5 kPa, and u_r(Na₂CO₃) = u_r(Na₂SO₄) = u_r(NaOH) = 0.05.

^bAbbreviations: S, Na₂SO₄; OH, NaOH; C₁, Na₂CO₃·H₂O; S1, Na₂SO₄·NaOH; and γ, mNa₂SO₄·nNa₂CO₃.

Figure 6.

Dry salt and water diagrams of the quaternary system NaOH–Na₂CO₃–Na₂SO₄–H₂O at 363.15 K and 88.94 kPa; (●) measured solubility; (—) solubility curve.

Figure 7.

X-ray diffraction photograph of the cosaturation points E₁, E₂, E₃, and E₄ of the quaternary system NaOH–Na₂CO₃–Na₂SO₄–H₂O at 363.15 K.

3.1.4. Na₂CO₃–NaOH–NaCl–H₂O System at 363.15 K

The experimental solubility data for the quaternary system Na₂CO₃–NaOH–NaCl–H₂O at 363.15 K were determined and combined with the literature data. These data are presented in Table 8 in 15 groups, and the solution composition of the equilibrium liquid phase is expressed in terms of the mass fraction and Jänecke index (g/100 g dry salt). The corresponding phase and water diagrams of the system are shown in Figure 8. The phase diagram consists of two points, E₁ and E₂, which are the cosaturation points of NaOH + Na₂CO₃ + NaCl and Na₂CO₃·H₂O + Na₂CO₃ + NaCl, respectively; and four crystallized regions, AGE₂FA, FBDE₁E₂F, GE₂E₁CG, and E₁DCH, which correspond to Na₂CO₃·H₂O, NaCl, Na₂CO₃, and NaOH, respectively. Among the crystallization fields, the crystallization field of Na₂CO₃·H₂O is the largest, suggesting that this salt is very easily crystallized because of its lower solubility in the quaternary system. Neither a solid solution nor a complex salt was formed. The XRD patterns of the cosaturation points (E₁ and E₂) are given in Figure 9.

Table 8. Solubilities of Solutions in the Quaternary System NaCl–Na₂CO₃–NaOH–H₂O at T = 363.15 K and P = 88.94 kPa^a.

	liquid mass %				dry salt (liquid) mass %
no.	NaOH	Na2CO3	NaCl	H2O	NaOH	Na2CO3	NaCl	H2O mass (g/100 g dry salt)	solution density ρ/g·cm–3	equilibrium phase solidsb
1	0.00	0.00	27.80	72.20	0.00	0.00	100.00	259.71	1.1723	Cl
2	68.87	0.00	2.87	28.26	96.00	0.00	4.00	39.39	1.5831	Cl + OH
3	76.28	0.00	0.00	23.72	100.00	0.00	0.00	31.10	1.5884	OH
4	48.79	0.55	0.00	50.66	98.88	1.12	0.00	102.68	1.4696	OH + C
5(E1)	51.00	2.05	4.01	42.94	89.38	3.59	7.03	75.26	1.5215	C + OH + Cl
6	36.51	1.95	3.64	57.90	86.72	4.64	8.64	137.52	1.3868	C + Cl
7(E2)	34.21	1.69	4.34	59.76	85.02	4.20	10.78	148.51	1.3680	C + C1 + Cl
8	17.14	10.99	0.00	71.87	60.93	39.07	0.00	255.49	1.2339	C + C1
9	0.00	30.50	0.00	69.50	0.00	100.00	0.00	227.87	1.3112	C1
10	0.00	9.97	21.09	68.94	0.00	32.10	67.90	221.96	1.2031	C1 + Cl
11	8.65	3.17	15.54	72.64	31.61	11.57	56.82	265.62	1.2084	C1 + Cl
12	16.42	2.57	10.71	70.30	55.28	8.67	36.05	236.73	1.2453	C1 + Cl
13	28.46	3.15	9.47	58.92	69.29	7.65	23.06	143.42	1.3418	C1 + Cl
14	24.67	1.58	7.14	66.61	73.88	4.73	21.39	199.44	1.2940	C1 + Cl
15	30.11	1.65	4.6	63.64	82.81	4.54	12.65	175.06	1.3286	C1 + Cl

^aStandard uncertainties u are u(T) = 0.1 K, u(P) = 0.5 kPa, and u_r(Na₂CO₃) = u_r(NaCl) = u_r(NaOH) = 0.05.

^bAbbreviations: OH, NaOH; Cl, NaCl; C, Na₂CO₃; and C₁, Na₂CO₃·H₂O.

Figure 8.

Equilibrium phase/water diagrams of the quaternary system NaCl–Na₂CO₃–NaOH–H₂O at 363.15 K and 88.94 kPa; (●) measured solubility; (—) solubility curve.

Figure 9.

X-ray diffraction photograph of the cosaturation point E₁ and E₂ of the quaternary system NaCl–Na₂CO₃–NaOH–H₂O at 363.15 K.

Figure 8 presents a superposition of the dry salt and water diagrams, which shows that the points and lines in the water diagram exactly correspond to those in the dry salt diagram.

3.1.5. NaOH–Na₂SO₄–NaCl–H₂O System at 363.15 K

The measured solubilities in the quaternary system NaOH–Na₂SO₄–NaCl–H₂O at 363.15 K are shown in Table 9 and are expressed in terms of the mass fraction and Jänecke index. By combining the abovementioned data, the dry salt and water diagrams of the system were prepared and are plotted in Figure 10. The system includes four three-salt cosaturation points (E₁, E₂, E₃, and E₄) and five crystallization fields (Na₂SO₄, NaCl, Na₂SO₄·NaCl·NaOH, Na₂SO₄·NaOH, and NaOH), among which the Na₂SO₄ and NaCl crystallization fields are larger than the other three. These results show that complex salts (S1, Na₂SO₄·NaOH and S3, Na₂SO₄·NaCl·NaOH) are formed in the system without the formation of a solid solution. Figure 11 shows the XRD patterns of the cosaturation points in this system. According to this figure, points E₁, E₂, E₃, and E₄ correspond to the cosaturation points of S1 + S3 + S, Cl + S + S3, OH + S3 + Cl, and S1 + S3 + OH, respectively.

Table 9. Experimental Values of Solubilities and Dry Salt of the Quaternary System NaOH–Na₂SO₄–NaCl–H₂O at T = 363.15 K and P = 88.94 kPa^a.

	liquid mass %				dry salt (liquid) mass %
no.	NaOH	Na2SO4	NaCl	H2O	NaOH	Na2SO4	NaCl	H2O mass (g/100 g dry salt)	solution density ρ/g·cm–3	equilibrium phase solidsb
1	76.28	0.00	0.00	23.72	100.00	0.00	0.00	31.10	1.5884	OH
2	71.62	2.75	0.00	25.63	96.30	3.70	0.00	34.46	1.5884	S1 + OH
3	34.18	2.22	0.00	63.60	93.90	6.10	0.00	174.73	1.3321	S1 + S
4(E1)	41.85	3.13	0.71	54.31	91.59	6.86	1.55	118.87	1.4190	S + S1 + S3
5	49.07	3.64	2.65	44.64	88.63	6.58	4.79	80.59	1.5030	S + S3
6(E2)	60.44	3.42	4.60	31.54	88.28	5.00	6.72	46.07	1.5831	S + Cl + S3
7	64.73	3.27	3.98	28.02	89.92	4.55	5.53	38.92	1.5831	Cl + S3
8	74.98	2.91	3.12	18.99	92.56	3.59	3.85	23.44	1.5831	Cl + S3
9(E3)	81.28	2.58	2.33	13.81	94.30	2.99	2.71	16.03	1.5832	Cl + S3 + OH
10	79.13	4.58	1.31	14.98	93.07	5.39	1.54	17.62	1.5831	S1 + S3
11(E4)	78.12	3.70	1.07	17.11	94.25	4.46	1.29	20.65	1.5831	S1 + S3 + OH
12	68.87	0.00	2.87	28.26	96.00	0.00	4.00	39.39	1.5831	Cl + OH
13	27.34	3.97	15.75	52.94	58.09	8.44	33.47	112.50	1.3485	S + Cl
14	13.80	4.24	19.17	62.79	37.08	11.40	51.52	168.75	1.2496	S + Cl
15	0.00	5.71	25.75	68.54	0.00	18.15	81.85	217.86	1.1768	S + Cl
16	0.00	0.00	27.80	72.20	0.00	0.00	100.00	259.71	1.1723	Cl
17	0.00	29.91	0.00	70.09	0.00	100.00	0.00	234.34	1.1984	S

^aStandard uncertainties u are u(T) = 0.1 K, u(P) = 0.5 kPa, and u_r(Na₂SO₄) = u_r(NaCl) = u_r(NaOH) = 0.05.

^bAbbreviations: Cl, NaCl; OH, NaOH; S, Na₂SO₄; S1, Na₂SO₄·NaOH; and S3, Na₂SO₄·NaCl·NaOH.

Figure 10.

Dry salt and water diagrams of the quaternary system NaOH–Na₂SO₄–NaCl–H₂O at 363.15 K and 88.94 kPa; (●) measured solubility; (—) solubility curve.

Figure 11.

X-ray diffraction photograph of the cosaturation points E₁, E₂, E₃, and E₄ of the quaternary system NaOH–Na₂SO₄–NaCl–H₂O at 363.15 K.

4. Solubility Predictions

For SLEs, Xu^1,19−21 proposed a new hypothesis for the reference state of activity coefficients with respect to solubility predictions. In the model, he assumed that the new reference state of activity coefficients is the activity of the solute: a_i (=m_i × γ_i) → 1 as m_i → the solubility. This reference is meaningful only for SLE, not other situations.

For the calculation, water–salt parameters for electrolyte solutions are shared in single salt systems and mixed salt systems. The excess Gibbs energy is described as follows

1

2

3

4

The function of G_w^e is based on ∑_i∑_j≠i(m_im_jλ_i,j) in the excess Gibbs energy function of the Pitzer model and G_w is adopted to account for the electrostatic interaction term between single electrolyte and water. The function of G_x^e is based on ∑_i∑_j,j≠i(m_im_jm_kλ_i,j,k) in Pitzer’s and G_x is the electrostatic interaction term between salt and salt, which is usually used in multicomponent electrolyte solutions. It is assumed that solubility of every dissolved solid is affected by other solutes in multicomponent electrolyte solutions, where m_i and m_j are the molalities of the solute; E_i, F_i,j, and G_i,j are the interaction parameters at a particular temperature; k_x = 0.5.

We obtain the activity coefficient expression of the solute from eqs 1–4, the expression for the solute is as follows

5

6

We obtain the activity expression from eq 6, and the activity of the solute is as follows

7

Then, based on the reference state of activity coefficients (a_i → 1 as m_i → the solubility) in this model, the solubility of the corresponding solute is obtained. The expression of the calculation is as follows

8

where m_x is the solubility of the solute.

The main expressions of the ion activity coefficient and the osmotic coefficient of water for the equation are briefly shown above. In this work, we calculated the solubility m_x of the corresponding solute. The software used for this calculation was 1stopt 6.0, and the computational algorithm was the Universal Global Algorithm.

Table 10 presents the values of the parameters introduced in eq 5; the solute–solute parameters¹ are listed.

Table 10. Salt–Salt Parameters for Electrolyte Solutions.

system	crystal	salts	pbcrystal-salt(1)	pbcrystal-salt(1)	pa(1)
NaCl–NaOH–H2O	NaCl	NaOH	–0.0149	0.4133	–2.1404
	NaOH	NaCl	0.7917	–0.4974	–4.3871
	NaOH	Na2SO4	–2.1514	2.0314	–4.3869
NaOH–Na2SO4–H2O	Na2SO4	NaOH	0.2092	–0.0141	–1.1481
	Na2SO4·NaOH	Na2SO4	0.0704	–1.0505	3.1256
	Na2SO4·NaOH	NaOH	–2.3238	–1.7023
	Na2CO3	Na2SO4	–1.9303	–6.1879	–1.0298
	Na2CO3	NaOH	0.1633	–0.1272
	Na2CO3·H2O	Na2SO4	–4.3639	3.2865	–1.2025
	Na2CO3·H2O	NaOH	–0.0428	0.3725
	Na2SO4	Na2CO3	–0.9579	0.9387	–1.1008
	Na2SO4	NaOH	0.1688	–0.1949
Na2CO3–Na2SO4–NaOH–H2O	NaOH	Na2CO3	2.1653	–0.6066	–4.2113
	NaOH	Na2SO4	–2.5062	2.7415
	mNa2SO4·nNa2CO3	Na2CO3	0.9340	–4.0026	6.2569
	mNa2SO4·nNa2CO3	Na2SO4	1.0652	–4.2955
	mNa2SO4·nNa2CO3	NaOH	–0.0042	0.0349
	NaCl	NaOH	3.2472	–3.7620	–1.7964
	NaCl	Na2CO3	–0.1235	1.1247
	NaOH	Na2CO3	–7.5784	6.1927	–4.3869
	Na2CO3	NaOH	0.1839	–0.0306
NaOH–Na2CO3–NaCl–H2O	Na2CO3	NaCl	–0.9737	8.2281	–14.6524
	Na2CO3·H2O	NaOH	4.8361	–6.4716
	Na2CO3·H2O	NaCl	–0.1109	0.5905	–1.2514
	Na2CO3·H2O		–1.0737	2.7899
	NaCl	NaOH	–0.7166	0.2424	–1.8233
	NaCl	Na2SO4	–0.0073	0.2111
	NaOH	Na2SO4	–2.1664	2.0445	–4.386
	NaOH	NaCl	–0.7109	1.1453
	Na2SO4	NaOH	–0.0667	0.7449	–0.6443
	Na2SO4	NaCl	0.8781	–1.9609
NaOH–Na2SO4–NaCl–H2O	Na2SO4·NaOH	NaOH	0.0159	–0.6345	1.2230
	Na2SO4·NaOH	Na2SO4	–1.4325	0.0923
	Na2SO4·NaOH	NaCl	2.1700	–1.6162
	Na2SO4·NaOH·NaCl	NaOH	0.0107	–0.4243	5.5244
	Na2SO4·NaOH·NaCl	Na2SO4	1.2604	–4.647
	Na2SO4·NaOH·NaCl	NaCl	1.1969	–4.7503

The predictive capability of the model was examined by calculating SLE in mixed electrolyte solution systems using the parameter values obtained from the correlations of binary electrolyte solution systems. The predicted SLE data for the solutions NaOH–NaCl–H₂O, NaOH–Na₂SO₄–H₂O, NaOH–Na₂CO₃–Na₂SO₄–H₂O, NaOH–Na₂CO₃–NaCl–H₂O, and NaOH–Na₂SO₄–NaCl–H₂O are shown in Tables 11–15 and Figures 12–16.

Table 11. Error Analysis for the NaOH–NaCl–H₂O System.

	experimental results, solubility/mol·kg–1 H2O				calculated results
T/K	NaCl	NaOH	solid phases		NaCl	NaOH	dPa/%
363.15	6.58	0.00	NaCl		6.78		3.02
	5.78	1.63	NaCl		5.46		5.58
	4.97	3.57	NaCl		4.91		1.18
	4.15	5.36	NaCl		3.91		5.74
	3.21	9.52	NaCl		3.01		6.24
	2.09	14.68	NaCl		2.17		3.85
	1.65	18.21	NaCl		1.73		4.64
	1.33	26.08	NaCl		1.38		3.37
	1.08	33.57	NaCl		1.07		0.61
	0.96	39.68	NaCl		0.96		0.20
	1.11	50.78	NaCl		1.23		10.81
	0.96	60.93	NaCl + NaOH		0.93	61.28	2.47	0.58
	0.79	63.72	NaOH			66.99	5.14
	0.84	68.49	NaOH			65.29	4.66
	0.61	73.87	NaOH			73.27	0.82
	0.00	80.39	NaOH			80.41	0.01
average							10.68	10.57

^adP = (1/N)∑|P_exp – P_cal|/P_exp × 100%.

Table 15. Error Analysis for the NaOH–Na₂SO₄–NaCl–H₂O System.

	experimental results, solubility/mol·kg–1 H2O				calculated results
T/K	NaOH	Na2SO4	NaCl	solid phasesb	NaOH	Na2SO4	NaCl	dPa/%
363.15	47.91	0.76	2.48	S + Cl + S3			2.73	10.25
	57.75	0.82	2.42	Cl + S3			2.69	11.22
	98.71	1.08	2.79	Cl + S3			2.71	3.14
	147.14	1.32	2.87	Cl + S3 + OH			2.72	5.36
	60.93	0.00	1.73	Cl + OH			1.82	5.38
	12.91	0.53	5.06	S + Cl			4.31	14.82
	5.49	0.48	5.19	S + Cl			4.86	6.32
	0.00	0.59	6.39	S + Cl			7.43	16.32
	0.00	0.00	6.55	Cl			6.32	3.53
	80.40	0.00	0.00	OH	80.40			0.00
	69.86	0.76	0.00	S1 + OH	69.86			0.00
	147.14	1.32	2.87	Cl + S3 + OH	147.14			0.00
	114.14	1.52	1.06	S1 + S3 + OH	114.14			0.00
	60.93	0.00	1.73	Cl + OH	60.93			0.00
	13.44	0.25	0.00	S1 + S		0.28		12.02
	19.26	0.41	0.22	S + S1 + S3		0.44		9.55
	27.48	0.57	1.01	S + S3		0.64		11.73
	47.91	0.76	2.48	S + Cl + S3		0.74		3.56
	12.91	0.53	5.06	S + Cl		0.45		15.17
	5.49	0.48	5.19	S + Cl		0.44		7.95
	0.00	0.59	6.39	S + Cl		0.60		2.42
	0.00	3.00	0.00	S		2.57		14.42
	69.86	0.76	0.00	S1 + OH	69.86	0.76		0.00
	13.44	0.25	0.00	S1 + S	13.44	0.25		0.00
	19.26	0.41	0.22	S + S1 + S3	19.26	0.41		0.00
	132.06	2.15	1.49	S1 + S3	132.06	2.15		0.00
	114.14	1.52	1.06	S1 + S3 + OH	114.14	1.52		0.00
	19.26	0.41	0.22	S + S1 + S3	19.27	0.41	0.22	0.04	0.00	0.00
	27.48	0.57	1.01	S + S3	27.43	0.57	1.01	0.19	0.00	0.00
	47.91	0.76	2.48	S + Cl + S3	47.88	0.76	2.48	0.05	0.00	0.00
	57.75	0.82	2.42	Cl + S3	58.04	0.83	2.43	0.49	1.22	0.41
	98.71	1.08	2.79	Cl + S3	98.24	1.07	2.78	0.47	0.94	0.36
	147.14	1.32	2.87	Cl + S3 + OH	147.35	1.32	2.87	0.14	0.00	0.00
	132.06	2.15	1.49	S1 + S3	132.04	2.15	1.49	0.01	0.00	0.00
	114.14	1.52	1.06	S1 + S3 + OH	114.20	1.52	1.06	0.05	0.00	0.00
average								4.42	4.44	4.40

^adP = (1/N)∑|P_exp – P_cal|/P_exp × 100%.

^bAbbreviations: Cl, NaCl; OH, NaOH; S, Na₂SO₄; S1, Na₂SO₄·NaOH; and S3, Na₂SO₄·NaCl·NaOH.

Figure 12.

Correlation of experimental SLE data for NaOH–NaCl–H₂O.

Figure 16.

Correlation of experimental SLE data for NaOH–Na₂SO₄–NaCl–H₂O.

Figure 13.

Correlation of experimental SLE data for NaOH–Na₂SO₄–H₂O.

Figure 14.

Correlation of experimental SLE data for Na₂CO₃–NaCl–NaOH–H₂O.

Figure 15.

Correlation of experimental SLE data for NaOH–Na₂CO₃–Na₂SO₄–H₂O.

Table 12. Error Analysis for the NaOH–Na₂SO₄–H₂O System.

	experimental results, solubility/mol·kg–1 H2O			calculated results
T/K	NaOH	Na2SO4	solid phases	NaOH	Na2SO4	dPa/%
363.15	0.00	3.00	Na2SO4		3.15	4.92
	2.29	2.28	Na2SO4		1.99	12.41
	3.76	1.57	Na2SO4		1.48	6.14
	5.08	1.20	Na2SO4		1.12	6.31
	5.84	0.93	Na2SO4		0.96	3.50
	7.13	0.58	Na2SO4		0.74	26.00
	9.14	0.50	Na2SO4		0.49	3.32
	11.24	0.26	Na2SO4		0.31	20.80
	13.44	0.25	Na2SO4 + Na2SO4·NaOH	13.41	0.25	0.16	0.00
	15.55	0.29	Na2SO4 + Na2SO4·NaOH	15.64	0.29	0.59	0.00
	15.74	0.27	Na2SO4 + Na2SO4·NaOH	15.71	0.27	0.20	0.00
	18.35	0.21	Na2SO4 + Na2SO4·NaOH	18.39	0.21	0.24	0.00
	20.24	0.29	Na2SO4 + Na2SO4·NaOH	20.12	0.29	0.57	0.00
	46.45	0.51	Na2SO4 + Na2SO4·NaOH	46.52	0.51	0.16	0.00
	69.86	0.76	Na2SO4 + Na2SO4·NaOH	69.82	0.75	0.76	0.06
	80.40	0.00	NaOH	80.40		0.00
average						5.38	5.22

^adP = (1/N)∑|P_exp – P_cal|/P_exp × 100%.

Table 13. Error Analysis for the NaOH–Na₂CO₃–NaCl–H₂O System.

	experimental results, solubility/mol·kg–1 H2O				calculated results
T/K	NaOH	Na2CO3	NaCl	solid phasesb	NaOH	Na2CO3	NaCl	dPa/%
363.15	0.00	0.00	6.55	Cl			6.03	7.95
	60.93	0.00	1.73	Cl + OH			1.72	0.65
	29.69	0.45	1.59	C + OH + Cl			1.49	6.44
	15.76	0.32	1.07	C + Cl			1.18	10.42
	14.31	0.27	1.24	C + C1 + Cl			1.47	19.03
	0.00	1.36	5.20	C1 + Cl			5.81	11.74
	2.98	0.41	3.64	C1 + Cl			3.67	0.90
	5.84	0.34	2.59	C1 + Cl			2.43	6.12
	12.08	0.50	2.73	C1 + Cl			2.49	8.85
	9.26	0.22	1.82	C1 + Cl			1.77	2.98
	11.83	0.24	1.23	C1 + Cl			1.29	4.98
	60.93	0.00	1.73	Cl + OH	60.93			0.00
	80.40	0.00	0.00	OH	80.40			0.00
	24.08	0.10	0.00	OH + C	24.08			0.00
	29.69	0.45	1.59	C + OH + Cl	29.69			0.00
	24.08	0.10	0.00	OH + C		0.10		0.00
	29.69	0.45	1.59	C + OH + Cl		0.45		0.00
	15.76	0.32	1.07	C + Cl		0.32		0.00
	14.31	0.27	1.24	C + C1 + Cl		0.27		0.00
	5.96	1.44	0.00	C + C1		1.44		0.00
	14.31	0.27	1.24	C + C1 + Cl		0.28		5.42
	5.96	1.44	0.00	C + C1		1.60		11.05
	0.00	4.14	0.00	C1		3.86		6.71
	0.00	1.36	5.20	C1 + Cl		1.45		6.51
	2.98	0.41	3.64	C1 + Cl		0.43		3.68
	5.84	0.34	2.59	C1 + Cl		0.32		6.93
	12.08	0.50	2.73	C1 + Cl		0.43		14.24
	9.26	0.22	1.82	C1 + Cl		0.27		18.51
	11.83	0.24	1.23	C1 + Cl		0.29		18.31
average								5.91

^adP = (1/N)∑|P_exp – P_cal|/P_exp × 100%.

^bAbbreviations: OH, NaOH; Cl, NaCl; C, Na₂CO₃; and C₁, Na₂CO₃·H₂O.

Table 14. Error Analysis for the NaOH–Na₂CO₃–Na₂SO₄–H₂O System.

	experimental results, solubility/mol·kg–1 H2O				calculated results
T/K	Na2CO3	Na2SO4	NaOH	solid phasesb	Na2CO3	Na2SO4	NaOH	dPa/%
363.15	0.00	3.00	0.00	S		3.00		0.00
	0.60	2.63	0.00	S + γ		2.58		1.88
	0.49	2.34	1.39	S + γ		2.48		6.03
	0.41	2.04	3.22	S + γ		2.01		1.20
	0.50	1.81	4.49	S + γ		1.77		2.07
	0.33	1.10	8.34	S + γ		1.03		5.77
	0.34	0.73	10.55	S + γ + S1		0.77		5.33
	1.01	0.00	69.86	S1 + OH			66.02	5.49
	0.00	0.00	80.40	OH			72.33	10.03
	0.10	0.00	24.08	C + OH			26.86	11.55
	0.49	0.29	34.59	OH + S1 + γ			33.47	3.23
	0.60	0.13	28.93	C1 + C + OH			30.84	6.58
	1.47	0.40	49.12	OH + C1 + γ			49.83	1.45
	0.10	0.00	24.08	C + OH	0.10			0.00
	1.44	0.00	5.96	C + C1	1.44			0.00
	0.60	0.13	28.93	C1 + C + OH	0.60			0.00
	1.44	0.00	5.96	C + C1	1.51			4.94
	4.14	0.00	0.00	C1	3.74			9.57
	2.67	0.62	0.00	C1 + γ	2.76			3.32
	1.67	0.40	49.12	OH + C1 + γ	1.46			0.63
	0.67	0.22	7.58	C1 + γ	0.74			9.07
	1.75	0.43	4.17	C1 + γ	1.54			12.10
	1.84	0.45	3.75	C1 + γ	1.72			6.74
	1.99	0.50	3.03	C1 + γ	1.89			4.68
	2.67	0.62	0.00	C1 + γ	2.69	0.62		0.71	0.00
	0.60	2.63	0.00	S + γ	0.59	2.61		0.84	0.76
	0.49	2.34	1.39	S + γ	0.49	2.32		0.64	0.86
	0.41	2.04	3.22	S + γ	0.41	2.07		1.57	1.47
	0.50	1.81	4.49	S + γ	0.50	1.81		0.26	0.00
	0.33	1.10	8.34	S + γ	0.33	1.11		1.60	0.91
	0.34	0.73	10.55	S + γ + S1	0.33	0.71		2.17	2.74
	0.49	0.29	34.59	OH + S1 + γ	0.48	0.29		1.64	0.00
	1.47	0.40	49.12	OH + C1 + γ	1.48	0.40		1.08	0.00
	0.67	0.22	7.58	C1 + γ	0.69	0.23		2.24	4.55
	1.75	0.43	4.17	C1 + γ	1.74	0.42		0.65	2.33
	1.84	0.45	3.75	C1 + γ	1.83	0.45		0.69	0.00
	1.99	0.50	3.03	C1 + γ	1.97	0.50		0.73	0.00
average								3.42	3.38

^adP = (1/N)∑|P_exp – P_cal|/P_exp × 100%.

^bAbbreviations: S, Na₂SO₄; OH, NaOH; C₁, Na₂CO₃·H₂O; S1, Na₂SO₄·NaOH; and γ, mNa₂SO₄·nNa₂CO₃.

The simulation data for the system at 363.15 K were obtained using the model and the parameters in Table 10. Comparisons of the experimental and calculated values of the ternary and quaternary invariant points for the quaternary system at 363.15 K are given in Tables 11–15; the calculated phase diagram is plotted using the simulation data in Figures 12–16. The analysis of the model as presented in the figures and tables shows that the simulation data agree well with the experimental data.

5. Conclusions

The solubilities of the quaternary systems (NaOH–Na₂CO₃–Na₂SO₄–H₂O, NaOH–Na₂CO₃–NaCl–H₂O, and NaOH–Na₂SO₄–NaCl–H₂O) at 363.15 K were determined using the isothermal solubility method. The solid minerals were identified using X-ray diffraction. According to the experimental data and the identification result, the isothermal phase diagrams were plotted. The phase diagrams determined for the quaternary systems were applied in the analysis of the vaporization process, laying the foundation for research on the phase equilibrium of the quinary system.

Combining the experimental solubility data of the systems, the corresponding parameters were fitted with Xu’s modified activity coefficient model. Then, the solubilities for the systems at 363.15 K were demonstrated. A comparison of the calculated and experimental solubilities of the systems showed that the predicted solubilities are in accordance with experimental values.

The authors declare no competing financial interest.