**4.1 Mineral identification for the solid phase**

For mineral identification when enough new solid phase appeared either in the stable equilibrium system or in the metastable equilibrium system, the wet residue mixtures were taken from the solution according to the experimental method. Firstly, as to the minerals of Na2B4O7·10H2O and K2B4O7·4H2O, the former belongs to monoclinic system, and the dual optical negative crystal i.e. 2ν(-) whereas the later belongs to trimetric system, and the dual optical positive crystal i.e. 2ν(+). Secondly, to the minerals NaCl and KCl, they can be identified through the property of refractive index. The refractive index of NaCl is higher than that of KCl. Observed with a XP-300D Digital Polarizing Microscopy using an oil immersion method, the crystal photos of the single and orthogonal polarized light on representative solid phases in the invariant points (NaCl + KCl + Na2B4O7·10H2O) and (Na2B4O7·10H2O + K2B4O7·4H2O + KCl) are presented in Figure 3.

single polarized light (10×10) orthogonal polarized light (10×10) (a) Invariant point (NaCl + KCl + Borax)

metastable evaporation, the solutions were not stirred, and the crystal behavior of solid phase was observed periodically. When enough new solid phase appeared, the wet residue mixtures were taken from the solution. The solids were then approximately evaluated by the combined chemical analysis, of XP-300D Digital Polarizing Microscopy (Shanghai Caikon Optical Instrument Co,. Ltd., China) using an oil immersion, and further identification with X-ray diffraction (X'pert PRO, Spectris. Pte. Ltd., The Netherlands). Meanwhile, a 5.0 cm3 sample of the clarified solution was taken from the liquid phase of each polyethylene container through a filter pipette, and then diluted to a 250.0 cm3 final volume in a volumetric flask filled with DDW for the quantitative analysis of the compositions of the liquid phase. Some other filtrates were used to measure the relative physicochemical properties individually according to the analytical method. The remainder of the solution continued to be evaporated and reached a new metastable

For mineral identification when enough new solid phase appeared either in the stable equilibrium system or in the metastable equilibrium system, the wet residue mixtures were taken from the solution according to the experimental method. Firstly, as to the minerals of Na2B4O7·10H2O and K2B4O7·4H2O, the former belongs to monoclinic system, and the dual optical negative crystal i.e. 2ν(-) whereas the later belongs to trimetric system, and the dual optical positive crystal i.e. 2ν(+). Secondly, to the minerals NaCl and KCl, they can be identified through the property of refractive index. The refractive index of NaCl is higher than that of KCl. Observed with a XP-300D Digital Polarizing Microscopy using an oil immersion method, the crystal photos of the single and orthogonal polarized light on representative solid phases in the invariant points (NaCl + KCl + Na2B4O7·10H2O) and

> single polarized light (10×10) orthogonal polarized light (10×10) (a) Invariant point (NaCl + KCl + Borax)

Borax

equilibrium.

**4. Experimental results** 

**4.1 Mineral identification for the solid phase** 

(Na2B4O7·10H2O + K2B4O7·4H2O + KCl) are presented in Figure 3.

Borax

KCl NaCl

single polarized light (10×10) orthogonal polarized light (10×10)

(b) Invariant point (KCl + Borax + K2B4O7·4H2O)

Fig. 3. Identification of the invariant points for the solid phase in the reciprocal system (NaCl - KCl – Na2B4O7 - K2B4O7 - H2O) with a polarized microscopy using an oil-immersion method. (a), the invariant point (NaCl + KCl + Na2B4O7·10H2O); (b), the invariant point (KCl + Na2B4O7·10H2O + K2B4O7·4H2O).

The metastable equilibria solid phases in the two invariant points are further confirmed with X-ray diffraction analysis, and listed in Figure 4, except in the invariant points (NaCl + KCl + Borax) in Figure 4a which shows that the minerals KCl, NaCl, Na2B4O7·10H2O and a minor Na2B4O7·5H2O are existed. The minor of Na2B4O7·5H2O maybe is formed due to the dehydration of Na2B4O7·10H2O in the processes of transfer operation and/or grinding.

(a), the X-ray diffraction photograph and the analytical data for the invariant point (NaCl + KCl + Borax)

Stable and Metastable Phase Equilibria in the Salt-Water Systems 409

100.00 100.00 0.00 0.00 100.00 96.79 85.85 0.00 11.98 24.64 41.25 96.67 90.05 90.59 94.45 93.36 92.13 93.83 95.55 96.67 98.30 96.19 96.05

Jänecke index

0.00 100.00 0.00 100.00 30.03 0.00 100.00 69.24 72.33 72.41 73.74 45.60 54.55 88.46 81.45 59.79 57.77 51.84 48.48 45.60 35.89 47.94 6.12

2- *J*(2Cl-) *J*(2K+) *J*(H2O)

*<sup>J</sup>*B, /[mol/100mol(2Na++2K+)] Equilirium

1807.5 2310.8 34623.3 7864.1 1526.2 1997.5 1807.8 5649.7 3947.5 3496.9 3181.6 1427.4 1672.8 2123.7 2191.0 1670.9 181.0 1602.8 1386.6 1427.4 1462.7 1406.5 1830.4

0 20 40 60 80 100

*J*(**2K+** )

**E**

**<sup>E</sup> <sup>F</sup> <sup>1</sup>**

**E4**

0

2000

**E 3**

4000

6000

solid phase\*\*

NaCl KCl N10 K4 NaCl+KCl NaCl+N10 KCl +K4 N10 + K4 N10 + K4 N10 + K4 N10 + K4 N10 + K4 KCl + N10 + K4 KCl+K4 KCl+K4 KCl+K4 KCl+K4 KCl + N10 KCl + N10 NaCl + KCl NaCl + KCl NaCl+KCl+N10 NaCl+N10

No.

1 2 3 4 5, E1 6, E2 7, E3 8, E4 9 10 11 12 13, E 14 15 16 17 18 19 20 21 22, F 23

10.25 0.00 0.72 0.00 8.02 9.60 0.00 1.15 1.38 1.57 1.59 3.50 4.72 1.13 1.57 4.18 4.15 5.27 6.11 6.40 7.50 6.16 9.73

Composition of the solution 100 *w*B\*

Na+ K+ Cl- B4O7

15.80 12.58 0.00 0.00 17.68 14.33 12.35 0.00 0.92 12.35 3.83 12.24 14.55 12.65 12.35 14.96 13.96 15.50 17.52 17.52 17.73 17.55 15.35

\* *w*B is in mass fraction; \*\* K4, K2B4O7·4H2O; N10, Na2B4O7·10H2O.

0 20 40 60 80 100

*J*( **2Cl- Na2B4O7** ) **2NaCl**

**Na2B4O7**·**10H2O**

**K2B4O7 2KCl**

**K2B4O7**·**4H2O**

0

20

40

*J*( **2K<sup>+</sup>** )

60

**E4**

80

100

Table 3. Stable solubilities of the system (NaCl - KCl – Na2B4O7 - K2B4O7 - H2O) at 298.15 K

**E F**

**KCl**

**E3**

**E1**

**NaCl**

 (a) dry-salt phase diagram (b) water-phase diagram Fig. 5. Stable phase diagram of the quaternary system (NaCl- KCl – Na2B4O7 - K2B4O7 - H2O)

0

2000

**E2**

4000

*<sup>J</sup>*(**H2 O**)

6000

**E 2**

at 298.15 K. (a), dry-salt phase diagram; (b), water-phase diagram.

0.00 0.00 2.41 9.42 0.00 1.04 4.48 12.54 14.77 1.59 11.97 7.62 3.17 2.78 1.59 2.33 2.61 2.58 1.79 1.32 0.67 1.52 1.35

0.00 13.87 0.00 4.74 5.85 0.00 15.87 4.36 6.11 11.74 7.55 11.39 9.62 13.44 11.74 10.56 9.65 9.43 9.83 9.10 7.13 9.64 1.05


(b), the X-ray diffraction photograph and the analytical data for the invariant point (KCl + Borax + K2B4O7·4H2O)

Fig. 4. The X-ray diffraction data of the invariant points. (a), the invariant point (NaCl + KCl + Na2B4O7·10H2O); (b), the invariant point (Na2B4O7·10H2O + K2B4O7·4H2O + KCl).

#### **4.2 Stable phase equilibrium of the quaternary system (NaCl - KCl – Na2B4O7 - K2B4O7 - H2O) at 298.15 K**

The stable phase equilibrium experimental results of solubilities of the quaternary system (NaCl - KCl – Na2B4O7 - K2B4O7 - H2O) at 298.15 K were determined, and are listed in Table 3, respectively. On the basis of the Jänecke index (*J*B, *J*B/[mol/100 mol(2Na+ + 2K+)]) in Table 3, the stable equilibrium phase diagram of the system at 298.15 K was plotted and shown in Figure 5.

#### **4.3 Metastable phase equilibrium of the quaternary system (NaCl - KCl – Na2B4O7 - K2B4O7 - H2O) at 308.15 K**

The experimental results of the metastable solubilities and the physicochemical properties of the quaternary system (NaCl - KCl – Na2B4O7 - K2B4O7 - H2O) at 308.15 K were determined, and are listed in Tables 4 and 5, respectively. On the basis of the Jänecke index (*J*B, *J*B/[mol/100 mol(2Na+ + 2K+)]) in Table 4, the metastable equilibrium phase diagram of the system at 308.15 K was plotted (Figure 6).


No. Visible Ref. Code Chemical Formula Score Scale

KCl

(b), the X-ray diffraction photograph and the analytical data for the invariant point

Fig. 4. The X-ray diffraction data of the invariant points. (a), the invariant point

K2(B4O5(OH)4)(H2O)2 B4O5(OH)4(Na2(H2O)8

(NaCl + KCl + Na2B4O7·10H2O); (b), the invariant point (Na2B4O7·10H2O + K2B4O7·4H2O +

The stable phase equilibrium experimental results of solubilities of the quaternary system (NaCl - KCl – Na2B4O7 - K2B4O7 - H2O) at 298.15 K were determined, and are listed in Table 3, respectively. On the basis of the Jänecke index (*J*B, *J*B/[mol/100 mol(2Na+ + 2K+)]) in Table 3, the stable equilibrium phase diagram of the system at 298.15 K was plotted and shown in

The experimental results of the metastable solubilities and the physicochemical properties of the quaternary system (NaCl - KCl – Na2B4O7 - K2B4O7 - H2O) at 308.15 K were determined, and are listed in Tables 4 and 5, respectively. On the basis of the Jänecke index (*J*B, *J*B/[mol/100 mol(2Na+ + 2K+)]) in Table 4, the metastable equilibrium phase diagram of the

01-072-1540 01-076-0753 01-074-0339

**4.2 Stable phase equilibrium of the quaternary system (NaCl - KCl – Na2B4O7 - K2B4O7 - H2O) at 298.15 K** 

**4.3 Metastable phase equilibrium of the quaternary system** 

**(NaCl - KCl – Na2B4O7 - K2B4O7 - H2O) at 308.15 K** 

system at 308.15 K was plotted (Figure 6).

1 2 3

KCl).

Figure 5.

True True True

(KCl + Borax + K2B4O7·4H2O)

Factor

0.650 0.011 0.004

39 37 10 Semi-Quant/%

72 22 5


\* *w*B is in mass fraction; \*\* K4, K2B4O7·4H2O; N10, Na2B4O7·10H2O.

Table 3. Stable solubilities of the system (NaCl - KCl – Na2B4O7 - K2B4O7 - H2O) at 298.15 K

(a) dry-salt phase diagram (b) water-phase diagram

Fig. 5. Stable phase diagram of the quaternary system (NaCl- KCl – Na2B4O7 - K2B4O7 - H2O) at 298.15 K. (a), dry-salt phase diagram; (b), water-phase diagram.

Stable and Metastable Phase Equilibria in the Salt-Water Systems 411

—\*\* — — — 5.63 7.29 7.72 7.81 — 8.22 8.66 9.21 8.51 9.55 9.43 — — 9.02 9.10 9.30 10.36 10.12 9.94 9.62 9.58 9.53 9.47 — 9.08 9.51 9.48 9.02 9.04 8.65 8.38

cm-3) pH Refractive index Viscosity

1.3405 1.3800 1.3742 1.3678 1.3869 1.3872 1.3880 1.3890 1.3802 1.3836 1.3860 1.3862 1.3879 1.3782 1.3798 — 1.3792 1.3814 1.3840 1.3863 1.3868 1.3868 1.3803 1.3823 1.3840 — 1.3856 — 1.3857 1.3842 1.3828 1.3827 1.3835 1.3841 1.3849 η/(mPa·s)

— — — — 1.1241 1.1452 1.1930 1.2620 — 1.3023 1.2865 1.2829 1.2729 0.8499 0.8443 — 0.8844 1.0625 1.0876 1.1963 2.8279 2.8032 1.5661 1.4548 — 1.3900 — — 1.1682 1.0812 1.0480 1.0252 1.0421 1.0809 1.0130

No.\* Density

ρ, /(g.

1.0441 1.1935 1.1857 1.2003 1.2300 1.2414 1.2433 1.2524 1.2060 1.2172 1.2274 1.2313 1.2398 1.2270 1.2347 — — 1.2433 1.2536 1.2700 1.3040 1.3206 1.2533 1.2600 1.2636 1.2764 1.2784 — 1.2555 1.2484 1.2409 1.2348 1.2366 1.2404 1.2381

\* Corresponding to the no. column in Table 4; \*\* not determined.

(NaCl - KCl – Na2B4O7 - K2B4O7 - H2O) at 308.15 K

Table 5. Physicochemical properties of the metastable reciprocal quaternary system


\* K4, K2B4O7·4H2O; N10, Na2B4O7·10H2O; *w*B, in mass fraction.

Table 4. Metastable solubilities of the quaternary system (NaCl - KCl – Na2B4O7 - K2B4O7 - H2O) at 308.15 K

0.00 100.00 100 0.00 100 98.54 97.6 94.98 96.47 95.63 95.33 95.22 94.92 85.66 85.99 86.66 87.60 85.56 81.57 77.34 0.00 15.03 39.58 49.49 62.43 65.66 72.68 75.94 80.09 85.45 87.64 89.98 90.62 91.80 93.91

Jänecke index

2- *J*(2Cl-) *J*(2K+) *J*(H2O)

0.00 0.00 100 100 32.94 32.89 33.09 32.89 0.00 8.25 15.27 18.97 27.96 100 94.61 94.16 86.86 80.30 73.39 71.90 71.66 75.41 76.36 79.22 79.81 79.97 76.57 72.72 69.15 67.31 63.8 60.17 57.36 50.79 43.95

*<sup>J</sup>*B, /[mol/100mol(2Na++2K+)] Equilibrium

2090.17 1788.31 18468.05 4311.59 1471.69 1469.10 1456.10 1445.77 1736.76 1684.84 1612.49 1566.69 1532.30 1955.76 1899.70 1894.74 1893.49 1844.98 1699.85 1641.11 2346.01 2134.79 2000.89 2160.56 1906.21 1773.28 1712.53 1838.60 1664.48 1723.59 1717.14 1699.65 1652.70 1604.93 1562.35

solid phase\*

N10 NaCl KCl K4 NaCl+KCl NaCl+KCl NaCl+KCl NaCl+KCl+N10 NaCl +N10 NaCl +N10 NaCl +N10 NaCl +N10 NaCl+N10 KCl+K4 KCl+K4 KCl+K4 KCl+K4 KCl+K4 KCl+K4 KCl+K4+N10 N10+K4 N10+K4 N10+K4 N10+K4 N10+K4 N10+K4 N10+K4 N10+K4 N10+KCl N10+KCl N10+KCl N10+KCl N10+KCl N10+KCl N10+KCl

No.

1.30 10.47 0.00 0.00 7.85 7.84 7.85 7.87 10.62 9.88 9.36 9.10 8.15 0.00 0.49 0.54 1.22 1.86 2.65 2.84 2.01 1.89 1.74 1.66 1.79 1.87 2.28 2.78 3.11 3.26 3.64 4.06 4.45 5.27 6.17

Composition of the solution 100*w*<sup>B</sup>

Na+ K+ Cl- B4O7

0.00 16.15 13.49 0.00 18.06 17.76 17.66 17.17 15.80 15.89 16.24 16.50 16.57 11.83 12.16 12.29 12.51 12.45 12.51 12.06 0.00 1.78 4.49 6.11 8.55 9.48 10.89 8.30 12.46 13.13 13.59 14.15 14.57 15.16 15.94

\* K4, K2B4O7·4H2O; N10, Na2B4O7·10H2O; *w*B, in mass fraction. Table 4. Metastable solubilities of the quaternary system (NaCl - KCl – Na2B4O7 - K2B4O7 - H2O) at 308.15 K

4.40 0.00 0.00 19.26 0.00 0.58 0.95 1.99 1.28 1.59 1.74 1.81 1.94 4.34 4.34 4.15 3.88 4.60 6.19 7.74 24.00 22.09 15.01 13.66 11.27 10.86 8.97 11.95 6.78 4.90 4.20 3.45 3.30 2.97 2.27

0.00 0.00 14.87 3.87 6.56 6.54 6.60 6.56 0.00 1.51 2.87 3.62 5.38 15.22 14.75 14.73 13.68 12.88 12.41 12.36 8.65 9.86 9.55 10.78 12.05 12.73 12.65 12.61 11.86 11.41 10.91 10.43 10.17 9.25 8.23


\* Corresponding to the no. column in Table 4; \*\* not determined.

Table 5. Physicochemical properties of the metastable reciprocal quaternary system (NaCl - KCl – Na2B4O7 - K2B4O7 - H2O) at 308.15 K

Stable and Metastable Phase Equilibria in the Salt-Water Systems 413

**0.0 0.5 1.0 1.5 2.0 2.5 3.0**

**E3**

**E2 E1 <sup>F</sup> E**

**0 5 10 15 20 25**

<sup>100</sup>*W*(**B4O7**

**F E**

**KCl**

**F'**

**E3**

**E'**

**E1**

**NaCl**

**E2**

**2-**)

**E4**

**E4**

**pH η/(mPa·s)** 

**0 5 10 15 20 25**

quaternary system (NaCl- KCl – Na2B4O7 - K2B4O7 - H2O) at 308.15 K

**2-**)

Fig. 7. Diagram of physicochemical properties versus composition for the metastable

**K2B4O7**·**4H2O**

**4.3 Comparison of the stable and metastable phase diagram of the quaternary system** 

A comparison of the dry-salt diagrams of the metastable phase equilibrium at 308.15 K and the stable phase equilibrium at 298.15 K for the same system is shown in Figure 8. The metastable crystallization regions of borax and potassium chloride are both enlarged while the crystallized area of other minerals existed is decreased. When compared with the stable system, the solubility of borax in water in the metastable system is increased from 3.13 % to 5.70 %. The metastable phenomenon of borax is obvious in this reciprocal quaternary

**K2B4O7 2KCl**

**0 20 40 60 80 100**

*J*( **2Cl- Na2B4O7** ) **2NaCl**

Fig. 8. Comparison of the metastable phase diagram at 308.15 K in solid line and the stable phase diagram at 298.15 K in dashed line for the quaternary system (NaCl - KCl – Na2B4O7 -

K2B4O7 - H2O). --, metastable experimental points; -○-, stable experimental points.

**Na2B4O7**·**10H2O**

**<sup>100</sup>***W*(**B4O7**

**E1**

**(NaCl - KCl – Na2B4O7 - K2B4O7 - H2O)** 

**0**

**20**

**40**

*J*( **2K<sup>+</sup>** )

**60**

**E4**

**80**

**100**

system.

**E3**

**F**

**E E2**

Fig. 6. Metastable phase diagram of the quaternary system (NaCl- KCl – Na2B4O7 - K2B4O7 - H2O) at 308.15 K. (a), dry-salt phase diagram; (b), water-phase diagram.

On the basis of physicochemical property data of the metastable system (NaCl - KCl – Na2B4O7 - K2B4O7 - H2O) at 308.15 K in Table 5, the diagram of physicochemical properties versus composition was drawn and shows in Figure 7. The physicochemical properties of the metastable equilibrium solution vary regularly with the composition of borate mass fraction. The singular point on every curve of the composition versus property diagram corresponds to the same invariant point and on the metastable solubility.

0 20 40 60 80 100 <sup>0</sup>

*J*( **2Cl-** ) **2NaCl Na2B4O7**

**Na2B4O7**·**10H2O**

**K2B4O7 2KCl**

**K2B4O7**·**4H2O**

**F'**

**E'**

**E'3**

**KCl**

**E'1**

**NaCl**

(a) dry-salt phase diagram (b) water-phase diagram

Fig. 6. Metastable phase diagram of the quaternary system (NaCl- KCl – Na2B4O7 - K2B4O7 -

On the basis of physicochemical property data of the metastable system (NaCl - KCl – Na2B4O7 - K2B4O7 - H2O) at 308.15 K in Table 5, the diagram of physicochemical properties versus composition was drawn and shows in Figure 7. The physicochemical properties of the metastable equilibrium solution vary regularly with the composition of borate mass fraction. The singular point on every curve of the composition versus property diagram

**E4**

**1.375**

**1.380**

**E2**

**1.385**

**E1**

**1.390** *n***<sup>35</sup>**

**E**

**F**

**0**

**2-**) **0 5 10 15 20 25**

**E3**

**<sup>W</sup>**(**B4O7**

**2-**)**,/%**

**E4**

**500**

**1000**

**1500**

**E'1**

**E'2**

**E'**

**E'3**

**F'**

**2000**

**2500** *<sup>J</sup>*( **H2O**)

**0 20 40 60 80 100**

**E'4**

*<sup>J</sup>*(**B4O7 2-** )

**E'2**

H2O) at 308.15 K. (a), dry-salt phase diagram; (b), water-phase diagram.

corresponds to the same invariant point and on the metastable solubility.

**0 5 10 15 20 25**

**<sup>100</sup>***W*(**B4O7**

**1.15**

**1.20**

**1.25**

**E1**

**E**

**E2 E3** **F**

*ρ***,/(g·cm-3)** 

**1.30**

**1.35**

20

40

*J*( **2K<sup>+</sup>** )

60

**E'4**

80

100

Fig. 7. Diagram of physicochemical properties versus composition for the metastable quaternary system (NaCl- KCl – Na2B4O7 - K2B4O7 - H2O) at 308.15 K

#### **4.3 Comparison of the stable and metastable phase diagram of the quaternary system (NaCl - KCl – Na2B4O7 - K2B4O7 - H2O)**

A comparison of the dry-salt diagrams of the metastable phase equilibrium at 308.15 K and the stable phase equilibrium at 298.15 K for the same system is shown in Figure 8. The metastable crystallization regions of borax and potassium chloride are both enlarged while the crystallized area of other minerals existed is decreased. When compared with the stable system, the solubility of borax in water in the metastable system is increased from 3.13 % to 5.70 %. The metastable phenomenon of borax is obvious in this reciprocal quaternary system.

Fig. 8. Comparison of the metastable phase diagram at 308.15 K in solid line and the stable phase diagram at 298.15 K in dashed line for the quaternary system (NaCl - KCl – Na2B4O7 - K2B4O7 - H2O). --, metastable experimental points; -○-, stable experimental points.

Stable and Metastable Phase Equilibria in the Salt-Water Systems 415

*φ* is the osmotic coefficient; *ZM* and *ZX* are the charges of anions and cautions in the solution. *m* is the molality of solute; υM, υX, and υ (υ = υM + υX) represent the stiochiometric

In equation (1), *fφ, B*φMX and CφMX are defined as following equations. In equation (1a), here *b*

*A I <sup>f</sup> bI* φ

*MX MX MX B e*

*MX MX MX MX B ee*

ββ

3 1000

π ρ<sup>=</sup>

β

For non 2-2 type of electrolytes, such as several 1-1-,2-1-, and 1-2-type pure salts, the best

φ

1 2 1 2 1 */ /*

*() () I* 0 1

1 2 *() () I () I* 01 2

3 2 1 2 <sup>2</sup> 1 2 <sup>0</sup>

*/ / N e <sup>W</sup> <sup>A</sup>*

*A<sup>φ</sup>* is the Debye-Hückel coefficient for the osmotic coefficient and equal to 0.3915 at 298.15 K. Where, *N*0 is Avogadro's number, *dw* and *D* are the density and static dielectric constant of the solvent (water in this case) at temperature and e is the electronic charge. *k* is Boltzmann's constant. In equation (1b), *β*(0) *MX*, *β*(1)*MX*, *C<sup>φ</sup>MX* are specific to the salt MX, and are the singleelectrolyte parameters of MX. The universal parameters α = 2.0 kg1/2·mol-1/2 and omit *β*(2)*MX* for several 1-1-,2-1-, and 1-2-type salts at 298.15 K. As salts of other valence types, the values α1 = 1.4 kg1/2·mol-1/2, and α2 = 12 kg1/2·mol-1/2 were satisfactory for all 2-2 or higher valence pairs electrolytes at 298.15 K. The parameter *β*(2)*MX* is negative and is related to the

*v v (v v ) ln z z f m B m <sup>C</sup>*

12 12 1 2 1 21 */ / / f A [I / ( bI ) ( / b)ln( bI )]*

γγ

γ

*MX MX MX B* =+ +

 βα  β

For 2-2 type of electrolytes, such as several 3-1- and even 4-1-type pure salts, an additional

 α

αα

*DkT*

3 2 2 2 <sup>2</sup> */ M X M X M X MX MX*

<sup>±</sup> =+ + (3)

= − ++ + (3a)

 α

<sup>2</sup> *g(x) [ ( x)ex* = −+ − 21 1 *p( x) / x ]* (3d)

*v v*

 φ

> β

0 1 12 2 12 *MX* 1 2 *() () / () /*

 β

− − =+ + (2c)

= − <sup>+</sup> (2a)

<sup>−</sup> = + (2b)

 γ

= + (3b)

*g(I ) g( I )]* (3c)

(2d)

coefficients of the anion, cation, and the total ions on the electrolyte MX.

φ

is a universal empirical constant to be equal 1.2 kg1/2·mol-1/2.

φ

φ

form of *B*<sup>φ</sup>*MX* is following (Pitzer, 1973):

term is added (Pizter, 1977):

association equilibrium constant.

The mean activity coefficient *γ<sup>±</sup>* is defined as:

γ

γ

 φ

β

*MX MX MX BBB*
