**5. Solubility theoretic prediction of salt-water system**

#### **5.1 Ion-interaction model**

As to any electrolyte, its thermodynamic prosperity varied from weak solution to high concentration could be calculated through 3 or 4 Pitzer parameters. Pitzer ion-interaction model and its extended HW model of aqueous electrolyte solution can be briefly introduced in the following (Pitzer, 1975, 1977, 2000; Harvie & Wear, 1980; Harvie et al., 1984; Kim & Frederich, 1988a-b).

As to the ion-interaction model, it is a semiempirical statistical thermodynamics model. In this model, the Pitzer approach begins with a virial expansion of the excess free energy of the form to consider the three kinds of existed potential energies on the ion-interaction potential energy in solution.

$$\mathbf{G}^{\rm ex} \mid (\mathbf{n}\_w \mathbf{R} \mathbf{T}) = f(I) + \sum\_{\mathbf{i}} \sum\_{\mathbf{j}} \lambda\_{\mathbf{i}\mathbf{j}}(I) m\_{\mathbf{i}} m\_{\mathbf{j}} + \sum\_{\mathbf{i}} \sum\_{\mathbf{j}} \sum\_{\mathbf{k}} \mu\_{\mathbf{i}\mathbf{j}\mathbf{k}} m\_{\mathbf{i}} m\_{\mathbf{j}} m\_{\mathbf{k}} + \dots \tag{1}$$

Where *n*w is kilograms of solvent (usually in water), and *m*i is the molality of species *i*  (species may be chosen to be ions); *i*, *j*, and *k* express the solute ions of all cations or anions; *I* is ion strength and given by

$$I = \frac{1}{2} \sum m\_i z\_i \, ^2 \text{, here z\'e is the number of charges on the } i\text{-th solute.}$$

The first term on the right in equation (1) is the first virial coefficients. The first virial coefficients i.e. the Debye-Hückel limiting law, *f*(*I*), is a function only of ionic strength to express the long-range ion-interaction potential energy of one pair of ions in solution and not on individual ionic molalities or other solute properties.

Short-range potential effects are accounted for by the parameterization and functionality of the second virial coefficients, *λ*ij, and the third virial coefficients, *μ*ijk. The quantity *λ*ij represents the short-range interaction in the presence of the solvent between solute particles *i* and *j*. This binary interaction parameter of the second virial coefficient does not itself have any composition dependence for neutral species, but for ions it is dependent it is ionic strength.

The quantity *μ*ijk represents short-range interaction of ion triplets and are important only at high concentration. The parameters *μ*ijk are assumed to by independent of ionic strength and are taken to be zero when the ions *i, j* and *k* are all cations or all anions.

Taking the derivatives of equation ? with respect to the number of moles of each components yields expressions for the osmotic and activity coefficients.

#### **5.1.1 For pure electrolytes**

For the pure single-electrolyte MX, the osmotic coefficient defined by Pitzer (2000):

$$\phi - 1 = \left| z\_M z\_X \right| f^\phi + m \frac{2v\_M v\_X}{v} B\_{MX}^\phi + m^2 \frac{2(v\_M v\_X)^{3/2}}{v} C\_{MX}^\phi \tag{2}$$

As to any electrolyte, its thermodynamic prosperity varied from weak solution to high concentration could be calculated through 3 or 4 Pitzer parameters. Pitzer ion-interaction model and its extended HW model of aqueous electrolyte solution can be briefly introduced in the following (Pitzer, 1975, 1977, 2000; Harvie & Wear, 1980; Harvie et al., 1984; Kim &

As to the ion-interaction model, it is a semiempirical statistical thermodynamics model. In this model, the Pitzer approach begins with a virial expansion of the excess free energy of the form to consider the three kinds of existed potential energies on the ion-interaction

ex ij i j ijk

Where *n*w is kilograms of solvent (usually in water), and *m*i is the molality of species *i*  (species may be chosen to be ions); *i*, *j*, and *k* express the solute ions of all cations or anions; *I*

<sup>2</sup> *i i I mz* <sup>=</sup> , here zi is the number of charges on the *i-*th solute.

The first term on the right in equation (1) is the first virial coefficients. The first virial coefficients i.e. the Debye-Hückel limiting law, *f*(*I*), is a function only of ionic strength to express the long-range ion-interaction potential energy of one pair of ions in solution and

Short-range potential effects are accounted for by the parameterization and functionality of the second virial coefficients, *λ*ij, and the third virial coefficients, *μ*ijk. The quantity *λ*ij represents the short-range interaction in the presence of the solvent between solute particles *i* and *j*. This binary interaction parameter of the second virial coefficient does not itself have any composition dependence for neutral species, but for ions it is dependent it is ionic

The quantity *μ*ijk represents short-range interaction of ion triplets and are important only at high concentration. The parameters *μ*ijk are assumed to by independent of ionic strength and

Taking the derivatives of equation ? with respect to the number of moles of each

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

*M X M X M X MX MX v v (v v ) zz f m B m <sup>C</sup>*

−= + + (2)

*v v*

*/*

 φ

For the pure single-electrolyte MX, the osmotic coefficient defined by Pitzer (2000):

φφ

*G / (n RT ) f (I ) <sup>w</sup> ijk* =+ + *<sup>λ</sup> (I )m m*

i j i jk

μ

*m m m .....* + (1)

**5. Solubility theoretic prediction of salt-water system** 

**5.1 Ion-interaction model** 

potential energy in solution.

is ion strength and given by

**5.1.1 For pure electrolytes** 

φ

strength.

1 <sup>2</sup>

not on individual ionic molalities or other solute properties.

are taken to be zero when the ions *i, j* and *k* are all cations or all anions.

components yields expressions for the osmotic and activity coefficients.

Frederich, 1988a-b).

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

In equation (1), *fφ, B*φMX and CφMX are defined as following equations. In equation (1a), here *b* is a universal empirical constant to be equal 1.2 kg1/2·mol-1/2.

$$f^{\phi} = -\frac{A^{\phi}I^{1/2}}{1 + bI^{1/2}}\tag{2a}$$

For non 2-2 type of electrolytes, such as several 1-1-,2-1-, and 1-2-type pure salts, the best form of *B*<sup>φ</sup>*MX* is following (Pitzer, 1973):

$$B\_{\rm MX}^{\phi} = \mathcal{B}\_{\rm MX}^{(0)} + \mathcal{B}\_{\rm MX}^{(1)} e^{-\alpha \sqrt{l}} \tag{2b}$$

For 2-2 type of electrolytes, such as several 3-1- and even 4-1-type pure salts, an additional term is added (Pizter, 1977):

$$B\_{\rm MX}^{\phi} = \beta\_{\rm MX}^{(0)} + \beta\_{\rm MX}^{(1)} e^{-\alpha\_1 \sqrt{l}} + \beta\_{\rm MX}^{(2)} e^{-\alpha\_2 \sqrt{l}} \tag{2c}$$

$$A^{\phi} = \frac{1}{3} \left(\frac{2\pi N\_0 \rho\_W}{1000}\right)^{1/2} \left(\frac{e^2}{DkT}\right)^{3/2} \tag{2d}$$

*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 association equilibrium constant.

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

$$\ln \gamma\_{\pm} = \left| z\_M z\_X \right| f^{\mathcal{I}} + m \frac{2v\_M v\_X}{\upsilon} B\_{MX}^{\mathcal{I}} + m^2 \frac{2(v\_M v\_X)^{3/2}}{\upsilon} C\_{MX}^{\mathcal{I}} \tag{3}$$

$$f^{\mathcal{I}} = -A^{\phi} \left[ \mathbf{l}^{1/2} \;/\; (1 + b \mathbf{l}^{1/2}) + (\mathbf{2} \;/\; b) \ln(\mathbf{1} + b \mathbf{l}^{1/2}) \right] \tag{3a}$$

$$B\_{MX}^{\mathcal{Y}} = B\_{MX} + B\_{MX}^{\mathcal{O}} \tag{3b}$$

$$B\_{\rm MX} = \beta\_{\rm MX}^{(0)} + \beta\_{\rm h\chi}^{(1)} \mathcal{g}(\alpha\_1 \mathbf{I}^{1/2}) + \beta\_{\rm MX}^{(2)} \mathcal{g}(\alpha\_2 \mathbf{I}^{1/2}) \,\tag{3c}$$

$$\mathbf{g(x) = 2\{1 - (1 + x)\exp(-x) / x^2\}}\tag{3d}$$

Stable and Metastable Phase Equilibria in the Salt-Water Systems 417

2. The single-electrolyte third virial coefficient, *C*MX, account for short-range interaction of ion triplets and are important only at high concentration. These terms are independent

> MX M X <sup>2</sup> */ C C /( Z Z )* φ

> > *i*

4. *A<sup>φ</sup>* is the Debye-Hückel coefficient for the osmotic coefficient and equal to 0.3915 at

parameters for each cation-cation-anion and anion-anion-cation triplet in mixed

ions represent measurable combinations of the second virial coefficients. They are defined as explicit functions of ionic strength by the following equations (Kim &

0 1 12 2 1 2

0 1 12 2 12

1 12 2 12

In Pitzer's model expression in Eqns. (10) to (12), is a function of electrolyte type and does not vary with concentration or temperature. Following Harvie et al. (1984), when either cation or anion for an electrolyte is univalent, the first two terms in equations (10) to (12) are

valence type, such as 2-2 electrolytes for these higher valence species accounts for their increased tendency to associate in solution, the full equations from (10) to (12) are used, and

can be neglect and α1 =2.0 kg1/2·mol-1/2, α2 = 0 at 298.15 K. For higher

 βα

 β

 βα

CA CA CA 1 2 CA *( ) ( ) /( ) / B exp( I ) exp( I)*

CA CA CA 1 CA 2 *() () / () / B* =+ +

CA CA 1 CA 2 *' () / () / B [* = +

 α

=+ − + −

298.15 K, and it is decided by solvent and temperature as equation (1d).

ψ

*C*φ

1 2

which describe the interaction of pair of oppositely charged

= (9)

*Z zm* <sup>=</sup> (10)

*i,j,k* in equations (3) to (5) are mixed electrolyte

 α

*g'( I ) g'( I )] / I* (13)

<sup>2</sup> *g(x) [ ( x)ex* = −+ − 21 1 *p( x)] / x* (14)

2 2 *g'(x) [ ( x x / )ex* =− − + + − 21 1 2 *p( x)] / x* (15)

*g(I ) g(I )* (12)

(11)

calculating the osmotic coefficient, are related by the equation (1-6) (Pitzer & Mayorga,

*i i*

, the corresponding coefficients for

of ionic strength. The parameters *<sup>C</sup>*MX and MX

MX

5. The third virial coefficients,

φ

φ

In equations (13) and (14), x = 1I1/2 or = 2I1/2.

α1=1.4 kg1/2·mol-1/2 and α2=12 kg1/2·mol-1/2 at 298.15 K.

ββ

β

 βα

βα

Where the functions g and g' in equations (10), (11) and (12) are defined by

electrolyte solutions. 6. The parameters *' B ,B, B CA CA*

Frederick, 1988).

considered, *( )* <sup>2</sup>

*CA* β

3. The function *Z* in the equation (8) is defined by:

Where, *m* is the molality of species *i*, and *z* is its charge.

1973):

$$\mathsf{C}\_{\text{MX}}^{\mathcal{I}} = \mathsf{BC}\_{\text{MX}}^{\emptyset} \;/\; \mathsf{2} \tag{3e}$$

#### **5.1.2 For mixture electrolytes**

In order to treat mixed electrolytes, the following sets of equations are identical with the form used by Harvie & Weare (1984) for modeling the osmotic coefficient and the activity coefficient of a neutral electrolyte based on Pitzer Equations.

$$\begin{split} \sum\_{i} m\_{i}(\boldsymbol{\phi} - \mathbf{1}) &= 2(-A^{\phi}I^{3/2} \;/\ (1 + 1.2I^{1/2}) + \sum\_{c=1}^{N\_{c}} \sum\_{a=1}^{N\_{a}} m\_{c} m\_{a}(B\_{\text{ca}}^{\phi} + \mathbf{Z}C\_{\text{ca}}) \\ &+ \sum\_{\mathbf{c}=1}^{N\_{c}-1} \sum\_{\mathbf{c'}=\mathbf{c+1}}^{N\_{c}} m\_{\mathbf{c}} m\_{\mathbf{c'}} (\Phi\_{\text{cc'}}^{\phi} + \sum\_{\mathbf{a}=1}^{N\_{\mathbf{a}}} m\_{\mathbf{a}} \boldsymbol{\mu}\_{\text{cc'}\mathbf{a}}) + \sum\_{\mathbf{a}=1}^{N\_{\mathbf{a}}-1} \sum\_{\mathbf{a}=\mathbf{a}+1}^{N\_{\mathbf{a}}} m\_{\mathbf{a}} m\_{\mathbf{a}} (\Phi\_{\text{aa'}}^{\phi} \\ &+ \sum\_{\mathbf{m}=1}^{N\_{\mathbf{a}}} m\_{\mathbf{c}} \boldsymbol{\mu}\_{\text{aa'}\mathbf{c}}) + \sum\_{\mathbf{a}=1}^{N\_{\mathbf{a}}} \sum\_{\mathbf{m}=\mathbf{a}}^{N\_{\mathbf{a}}} m\_{\mathbf{m}} m\_{\mathbf{a}} \lambda\_{\mathbf{m}} \end{split} \tag{4}$$

$$\begin{aligned} \mathop{\rm c=1}{\mathop{\rm c=1}{\mathop{\rm c=1}{\mathop{\rm c=1}{\mathop{\rm c=1}{\mathop{\rm c}{\mathop{\rm c}}{\mathop{\rm c}}}}} & \mathop{\rm n=1}{\mathop{\rm c=1}{\mathop{\rm c}}} & \mathop{\rm n=1}{\mathop{\rm c}} & \mathop{\rm n=1}{\mathop{\rm c}} \\ \frac{1}{\mathop{\rm c}} \sum\_{\mathbf{a}=1}^{N\_{\mathop{\rm a}}} m\_{\mathbf{a}} (2B\_{\mathbf{Ma}} + \mathbf{ZC\_{\mathbf{Ma}}}) + \sum\_{\mathbf{c}=1}^{N\_{\mathop{\rm c}}} m\_{\mathbf{c}} (2\Phi\_{\mathbf{Ma}} + \sum\_{\mathbf{a}=1}^{N\_{\mathop{\rm a}}} m\_{\mathbf{a}} \mathsf{w}\_{\mathbf{Ma}}) + \\ \quad \sum\_{\mathbf{a}=1}^{N\_{\mathop{\rm a}}-1} \sum\_{\mathbf{a}'=\mathbf{a}+1}^{N\_{\mathop{\rm a}}} m\_{\mathbf{a}} m\_{\mathbf{a}} \mathsf{w}\_{\mathbf{a}} \mathsf{w}\_{\mathbf{Ma}} + \Big| \mathop{\rm c}\_{\mathop{\rm c}\mathbf{M}} \Big| \sum\_{\mathbf{c}=1}^{N\_{\mathop{\rm c}}} \sum\_{\mathbf{a}=1}^{N\_{\mathop{\rm a}}} m\_{\mathbf{c}} m\_{\mathbf{a}} \mathsf{C\_{\mathbf{ca}}} + \sum\_{\mathbf{n}=1}^{N\_{\mathop{\rm n}}} m\_{\mathbf{n}} (\mathsf{LA}\_{\mathbf{nM}}) \end{aligned} \tag{5}$$

$$\begin{aligned} \ln \gamma\_{\text{X}} = z\_{\text{X}} \,^2 F + \sum\_{\text{c}=1}^{N\_{\text{c}}} m\_{\text{c}} (2B\_{\text{c}\text{X}} + ZC\_{\text{c}\text{X}}) + \sum\_{\text{a}=1}^{N\_{\text{a}}} m\_{\text{a}} (2\Phi\_{\text{Xa}} + \sum\_{\text{c}=1}^{N\_{\text{c}}} m\_{\text{c}} \mathcal{V}\_{\text{Xa}\text{c}}) + \\ \sum\_{\text{c}=1}^{N\_{\text{c}}-1} \sum\_{\text{c}'=\text{c}+1}^{N\_{\text{c}}} m\_{\text{c}} m\_{\text{c}} \mathcal{V}\_{\text{c}\text{X}} + \left| z\_{\text{X}} \right| \sum\_{\text{c}=1}^{N\_{\text{c}}} \sum\_{\text{a}=1}^{N\_{\text{a}}} m\_{\text{c}} m\_{\text{a}} \mathcal{C}\_{\text{ca}} + \sum\_{\text{n}=1}^{N\_{\text{n}}} m\_{\text{n}} (2\mathcal{A}\_{\text{n}\text{X}}) \\ \ln \gamma\_{\text{N}} = \sum\_{\text{c}=1}^{N\_{\text{c}}} m\_{\text{c}} (2\mathcal{A}\_{\text{nc}}) + \sum\_{\text{a}=1}^{N\_{\text{a}}} m\_{\text{a}} (2\mathcal{A}\_{\text{na}}) \end{aligned} \tag{7}$$

In equations (3), (4), (5) and (6), the subscripts M, c, and c' present cations different cations; X, a, and a' express anions in mixture solution. *N*c, *N*a and *N*n express the numbers of cations, anions, and neutral molecules; *r*M, *Z*M, *m*C and *r*X, *Z*X, *m*a, *Ф* present the ion activity coefficient, ion valence number, ion morality, and the permeability coefficient; *γ*n, *m*n, *λ*nc, and *λ*na express activity coefficient of neutral molecule, morality of neutral molecule the interaction coefficient between neutral molecules with cations c and anion a.

In equations from (3) to (6), the function symbols of *FCZA B B* ,,, , ,, , φ φ ψ Φ are as following, respectively:

1. The term of *F* in equations (4) to (5) depends only on ionic strength and temperature. The defining equation of *F* is given by equation (7).

$$\begin{aligned} F &= -A^{\theta} \left[ 1^{1/2} \;/\, (1 + 1.21^{1/2}) + 2 \;/\, 1.2 \ln(1 + 1.21^{1/2}) \right] + \sum\_{\mathbf{c} = 1}^{N\_{\mathbf{c}}} \sum\_{\mathbf{a} = 1}^{N\_{\mathbf{a}}} m\_{\mathbf{c}} m\_{\mathbf{a}} B\_{\mathbf{c}}^{'} \\ &+ \sum\_{\mathbf{c} = 1}^{N\_{\mathbf{c}} - 1} \sum\_{\mathbf{c'} = \mathbf{c} + 1}^{N\_{\mathbf{c}}} m\_{\mathbf{c}} m\_{\mathbf{c'}} \Phi\_{\mathbf{c'}}^{'} + \sum\_{\mathbf{a} = 1}^{N\_{\mathbf{a}} - 1} \sum\_{\mathbf{a} = \mathbf{a} + 1}^{N\_{\mathbf{a}}} m\_{\mathbf{a}} m\_{\mathbf{a}} \Phi\_{\mathbf{a} \mathbf{c'}}^{'} \end{aligned} \tag{8}$$

 φ

In order to treat mixed electrolytes, the following sets of equations are identical with the form used by Harvie & Weare (1984) for modeling the osmotic coefficient and the activity

c c a a a

c=1 c c+1 a=1 a=1 a' a 1

 λ

c

= = +

+ Φ+ + Φ

1 1

*N N N NN*

− −

a c a

= + + + Φ+ +

*N N N*

M M a Ma Ma c Mc a Mca a=1 c=1 a=1

*ln z F m ( B ZC ) m ( m )*

ψ

2 2

+ =

X X c cX cX a Xa c Xac c=1 a=1 c=1

*ln z F m ( B ZC ) m ( m )*

ψ

γλ

interaction coefficient between neutral molecules with cations c and anion a.

In equations from (3) to (6), the function symbols of *FCZA B B* ,,, , ,, ,

c c a a

c=1 c'=c+1 a=1 a'=a+1

1 1

*N N N N*

− −

The defining equation of *F* is given by equation (7).

2 2

a a c a n

*N N N N N*

a=1 a'=a 1 c=1 a 1 n=1

c c c a n

*N N N N N*

c=1 c'=c+1 c=1 a=1 n=1

2 2

c a c

= + + + Φ+ +

c a N c nc a na c=1 a=1

In equations (3), (4), (5) and (6), the subscripts M, c, and c' present cations different cations; X, a, and a' express anions in mixture solution. *N*c, *N*a and *N*n express the numbers of cations, anions, and neutral molecules; *r*M, *Z*M, *m*C and *r*X, *Z*X, *m*a, *Ф* present the ion activity coefficient, ion valence number, ion morality, and the permeability coefficient; *γ*n, *m*n, *λ*nc, and *λ*na express activity coefficient of neutral molecule, morality of neutral molecule the

1. The term of *F* in equations (4) to (5) depends only on ionic strength and temperature.

cc' aa'

*' '*

c c' a a'

*m m m m*

1 12 2 12 1 12

= − ++ + +

+ Φ + Φ

*F A [ I / ( . I ) / . ln( . I )] m m B*

1 2 1 2 1 2 c a

*/ / / '*

*N N ln m ( ) m ( )*

*N NN*

*m ( ) ( A I / ( . I ) m m (B ZC )*

−=− + + +

φ

c aa'c n cn

*N N* 

n=1 c=1

*m ) mm*

*/ /*

c a

*mm ( m ) mm (*

ψ

*N N*

c c' cc' a cc'a a a' aa'

a a' aa'M M c a ca n nM

c c' cc'X X c a ca n nX

 λ= + (7)

φ

ψ

 φ

c a

*N N*

c=1 a=1

Φ are as following,

ca

(8)

*mm z mmC m ( )*

+ +

*mm z mmC m ( )*

+ +

3 2 1 2 c a ca ca =1 =1

= (3e)

 φ

> ψ

 ψ

 φ

2

2

 λ

 λ (4)

(5)

(6)

*C C/ MX MX* 3 2

γ

coefficient of a neutral electrolyte based on Pitzer Equations.

φ

1 2 1 12

*'*

ψ

<sup>n</sup> <sup>c</sup>

+ +

*i c a*

c

*N*

c=1

1

1

−

−

2

2

**5.1.2 For mixture electrolytes** 

*i*

φ

γ

respectively:

φ

γ

2. The single-electrolyte third virial coefficient, *C*MX, account for short-range interaction of ion triplets and are important only at high concentration. These terms are independent of ionic strength. The parameters *<sup>C</sup>*MX and MX *C*φ , the corresponding coefficients for calculating the osmotic coefficient, are related by the equation (1-6) (Pitzer & Mayorga, 1973):

$$\mathbf{C}\_{\text{MX}} = \mathbf{C}\_{\text{acc}}^{\phi} \;/\ (2 \left| \mathbf{Z}\_{\text{M}} \mathbf{Z}\_{\text{X}} \right|^{1/2}) \tag{9}$$

3. The function *Z* in the equation (8) is defined by:

$$Z = \sum\_{i} |z\_i| m\_i \tag{10}$$

Where, *m* is the molality of species *i*, and *z* is its charge.


$$B\_{\rm CA}^{\phi} = \beta\_{\rm CA}^{(0)} + \beta\_{\rm CA}^{(1)} \exp(-\alpha\_1 I^{1/2}) + \beta\_{\rm CA}^{(2)} \exp(-\alpha\_2 I^{1/2}) \tag{11}$$

$$B\_{\rm CA} = \beta\_{\rm CA}^{(0)} + \beta\_{\rm CA}^{(1)} \lg(\alpha\_1 \mathbf{I}^{1/2}) + \beta\_{\rm CA}^{(2)} \lg(\alpha\_2 \mathbf{I}^{1/2}) \tag{12}$$

$$B\_{\rm CA}^{\rm \cdot} = \lceil \beta\_{\rm CA}^{\prime \rm 1} g^{\prime}(\alpha\_1 \mathbf{I}^{1/2}) + \beta\_{\rm CA}^{\prime \rm 2} g^{\prime}(\alpha\_2 \mathbf{I}^{1/2}) \rceil / \,\,\mathrm{I} \tag{13}$$

Where the functions g and g' in equations (10), (11) and (12) are defined by

$$\mathbf{g(x) = 2[1 - (1 + x)\exp(-x)] / x^2} \tag{14}$$

$$\text{g'} (\text{x}) = -2 \left[ 1 - (1 + \text{x} + \text{x}^2 \text{ / 2}) \exp(-\text{x}) \right] / \text{x}^2 \tag{15}$$

In equations (13) and (14), x = 1I1/2 or = 2I1/2.

In Pitzer's model expression in Eqns. (10) to (12), is a function of electrolyte type and does not vary with concentration or temperature. Following Harvie et al. (1984), when either cation or anion for an electrolyte is univalent, the first two terms in equations (10) to (12) are considered, *( )* <sup>2</sup> *CA* β can be neglect and α1 =2.0 kg1/2·mol-1/2, α2 = 0 at 298.15 K. For higher valence type, such as 2-2 electrolytes for these higher valence species accounts for their increased tendency to associate in solution, the full equations from (10) to (12) are used, and α1=1.4 kg1/2·mol-1/2 and α2=12 kg1/2·mol-1/2 at 298.15 K.

Stable and Metastable Phase Equilibria in the Salt-Water Systems 419

systems, and even geological fluids (Felmy & Weare, 1986; Kim & Frederich, 1988a, 1988b;

By the way, additional work has centered on developing variable-temperature models, which will increase the applicability to a number of diverse geochemical systems. The primary focus has been to broaden the models by generating parameters at higher or lower temperatures (Pabalan & Pitzer, 1987; Spencer et al., 1990; Greenberg & Moller, 1989).

As to the borate solution, the crystallized behavior of borate salts is very complex. The coexisted polyanion species of borate in the liquid phase is difference with the differences of boron concentration, pH value, solvent, and the positively charged ions. The ion of B4O72- is the general statistical express for various possible existed borates. Therefore, the structural formulas of Na2B4O7·10H2O and K2B4O7·4H2O in the solid phases of the quaternary system (NaCl - KCl – Na2B4O7 - K2B4O7 - H2O) are Na2[B4O5(OH)4]·8H2O and K2[B4O5(OH)4]·2H2O, respectively. Borate in the liquid phase corresponding to the equilibrium solid phase maybe

Therefore, in this part of predictive solubility of the quaternary system (NaCl - KCl – Na2B4O7 - K2B4O7 - H2O), the predictive solubilities of this system were calculated on the basis of two assumptions: Model I: borate in the liquid phase exists all in statistical form of

The necessary model parameters for the activity coefficients of electrolytes in the system at 298.15 K were fit from obtained osmotic coefficients and the sub-ternary subsystems by the

Model I: Suppose that borate in solution exists as in the statistical expression form of B4O72-

2-, and the dissolved equilibria in the system could be following:

2 8 *K (m ) (m ) a <sup>N</sup>*<sup>10</sup> *Na Na BB w* 4 4 = ⋅ ⋅ ⋅⋅ + + γ

 2 2 *K (m ) (m ) a <sup>K</sup>*<sup>4</sup> *K K BB w* 4 4 = ⋅ ⋅ ⋅⋅ + + γ

*NaCl Na Na Cl Cl K (m ) (m )* = ⋅⋅⋅ + + −− γ

 γ

 γ

> γ

(24)

(26)

(25)

2-, B3O3(OH)4-, B(OH)4-, and son on due to the reactions of

2-; Model II: borate in the liquid phase exists as various boron species of

Fang et al., 1993; Song, 1998; Song & Yao, 2001, 2003; Yang, 1988, 1989, 1992, 2005).

**5.2 Model parameterization and solubility predictions** 

polymerization or depolymerization of boron anion.

multiple and unary linear regression methods.

**5.2.1 Model I for the solubility prediction** 

.

Na2B4O7·10H2O = 2Na+ + B4O5(OH)42- + 8H2O

K2B4O7·4H2O = 2K+ + B4O5(OH)42- + 2H2O

So, the dissolved equilibrium constants can be expressed as:

coexists as B4O5(OH)4

B4O72- i.e. B4O5(OH)4

i.e. B4O5(OH)4

B4O5(OH)42-, B3O3(OH)4-, B(OH)4-

NaCl = Na+ + Cl-

KCl = K+ + Cl-

7. ij ij ij *' , ,* φ ΦΦΦ which depend upon ionic strength, are the second virial coefficients, and are given the following form (Pitzer, 1973).

$$\boldsymbol{\Phi}\_{\rm ij}^{\phi} = \boldsymbol{\theta}\_{\rm ij} + \prescript{E}{}{\boldsymbol{\Theta}}\_{\rm ij} + \prescript{E}{}{\boldsymbol{\Theta}}\_{\rm ij}^{\cdot} \tag{16}$$

$$
\Phi\_{\rm ii} = \Theta\_{\rm ii} + \prescript{E}{}{\Theta}\_{\rm ii} \tag{17}
$$

$$
\boldsymbol{\dot{\Phi}}\_{\text{ij}}^{\dot{\cdot}} = \,^{E}\boldsymbol{\dot{\theta}}\_{\text{ij}}^{\dot{\cdot}} \tag{18}
$$

In equations (15), (16) and (17), θ*i,j* is an adjustable parameter for each pair of anions or cations for each cation-cation and anion-anion pair, called triplet-ion-interaction parameter. The functions, ij *E* θ and ' ij *E* θ are functions only of ionic strength and the electrolyte pair type. Pitzer (1975) derived equations for calculating these effects, and Harvie and Weare (1981) summarized Pitzer's equations in a convenient form as following:

$$\:^E\Theta\_{\vec{\mathbb{I}}\vec{\mathbb{I}}} = (Z\_{\vec{\mathbb{I}}}Z\_{\vec{\mathbb{I}}} \, \, \, 4\, \mathbf{I}) \{ \, \mathbf{J}(\mathbf{x}\_{\vec{\mathbb{I}}\vec{\mathbb{I}}}) - \mathbf{J}(\mathbf{x}\_{\vec{\mathbb{I}}\vec{\mathbb{I}}}) \, \, \, \, \, \mathbf{2} - \mathbf{J}(\mathbf{x}\_{\vec{\mathbb{I}}\vec{\mathbb{I}}}) \, \, \, \, \mathbf{2} \} \tag{19}$$

$$\mathbf{^Eq}\_{\mathbf{i}\mathbf{j}} = -(\mathbf{^Eq}\_{\mathbf{i}\mathbf{j}} \;/\; \mathbf{I}) + (\mathbf{Z\_i}\mathbf{Z\_j} \;/\; 8\mathbf{I}^2) \mathbf{(x\_{i\mathbf{j}})} \mathbf{(x\_{i\mathbf{j}})} - \mathbf{x\_{i\mathbf{i}}} \mathbf{J'} (\mathbf{x\_{i\mathbf{i}}}) \;/\; \mathbf{2} - \mathbf{x\_{j\mathbf{j}}} \mathbf{J'} (\mathbf{x\_{j\mathbf{j}}}) /\; \mathbf{2} \tag{20}$$

$$\alpha\_{\rm ij} = 6Z\_{\rm i}Z\_{\rm j}A^{\phi}I^{1/2} \tag{21}$$

In equations (18) and (19), *J(x)* is the group integral of the short-range interaction potential energy. *J'(x)* is the single-order differential quotient of *J(x)* , and both are independent of ionic strength and ion charges. In order to give the accuracy in computation, *J(x)* can be fitted as the following function:

$$f(\mathbf{x}) = 1 \;/\; 4\mathbf{x} - \mathbf{1} + 1 \;/\; \mathbf{x} \Big|\_{0}^{\prime \infty} \left[1 - \exp(-\mathbf{x} \;/\; y \mathbf{C}\_1 \mathbf{x}^{-\mathbf{C}\_2} \cdot \exp(-\mathbf{C}\_3 \mathbf{x}^{\mathbf{C}\_4})) \right]^{-1} \tag{22}$$

$$\begin{split} \mathbf{J}'(\mathbf{x}) &= \left[ 4 + \mathbf{C}\_1 \mathbf{x}^{-\mathbf{C}\_2} \cdot \exp(-\mathbf{C}\_3 \mathbf{x}^{\mathbf{C}\_4}) \right]^{-1} \\ &+ \left[ 4 + \mathbf{C}\_1 \mathbf{x}^{-\mathbf{C}\_2} \exp(-\mathbf{C}\_3 \mathbf{x}^{\mathbf{C}\_4}) \right]^{-2} \left[ \mathbf{C}\_1 \mathbf{x} \exp(-\mathbf{C}\_3 \mathbf{x}^{\mathbf{C}\_4}) (\mathbf{C}\_2 \mathbf{x}^{-\mathbf{C}\_2 - 1} + \mathbf{C}\_3 \mathbf{C}\_4 \mathbf{x}^{\mathbf{C}\_4 - 1} \mathbf{x}^{-\mathbf{C}\_2}) \right] \end{split} \tag{23}$$

In equations (21) and (22), *C*1 = 4.581, *C*2 = 0.7237, *C*3 = 0.0120, *C*4 = 0.528.

Firstly, *x*ij can be calculated according to equation (20), and *J(x)* and *J'(x)* were obtained from equations (21) and (22), and then to obtained ij *E* θ and ' ij *E* θ from equations (18) and (19); finally, ij ij ij *' , ,* φ ΦΦΦ can be got through equations from (15) to (17). Using the values of ij ij ij *' , ,* φ ΦΦΦ , the osmotic and activity coefficients of electrolytes can be calculated via equations from (3) to (6).

Using the osmotic coefficient, activity coefficient and the solubility products of the equilibrium solid phases allowed us to identify the coexisting solid phases and their compositions at equilibrium.

On Pitzer ion-interaction model and its extended HW model, a numbers of papers were successfully utilized to predict the solubility behaviors of natural water systems, salt-water

ij ij ij ij *E EI*

ij ij ij *<sup>E</sup>* Φ= + θ

> ' ' ij ij *<sup>E</sup>* Φ = θ

cations for each cation-cation and anion-anion pair, called triplet-ion-interaction parameter.

Pitzer (1975) derived equations for calculating these effects, and Harvie and Weare (1981)

ij i j ij ii jj 4 22 *<sup>E</sup>*

ij ij i j ij ij ii ii jj jj <sup>8</sup> 2 2 *E E*

ij i j <sup>6</sup> */ x ZZA I* φ

In equations (18) and (19), *J(x)* is the group integral of the short-range interaction potential energy. *J'(x)* is the single-order differential quotient of *J(x)* , and both are independent of ionic strength and ion charges. In order to give the accuracy in computation, *J(x)* can be

1 3 1 32 3 4

Firstly, *x*ij can be calculated according to equation (20), and *J(x)* and *J'(x)* were obtained

ΦΦΦ , the osmotic and activity coefficients of electrolytes can be calculated via

Using the osmotic coefficient, activity coefficient and the solubility products of the equilibrium solid phases allowed us to identify the coexisting solid phases and their

On Pitzer ion-interaction model and its extended HW model, a numbers of papers were successfully utilized to predict the solubility behaviors of natural water systems, salt-water

*[ C x exp( C x )] [C x exp( C x )(C x C C x x )]*

 θ

 Φ= + + θθ

φ

θ

summarized Pitzer's equations in a convenient form as following:

4

In equations (21) and (22), *C*1 = 4.581, *C*2 = 0.7237, *C*3 = 0.0120, *C*4 = 0.528.

++ − − +

2 1

*C C*

− −

1 3

from equations (21) and (22), and then to obtained ij

*J'(x) [ C x exp( C x )]*

=+ ⋅ −

ΦΦΦ which depend upon ionic strength, are the second virial coefficients, and are

'

*i,j* is an adjustable parameter for each pair of anions or

= (21)

2 4 1

from equations (18) and (19);

are functions only of ionic strength and the electrolyte pair type.

= −− *(Z Z / I )[ J(x ) J(x ) / J(x ) / ]* (19)

=− + *( / I ) (Z Z / I )[x J'(x ) x J'(x ) / x J'(x ) / ]* − − (20)

1 2

1 3 <sup>0</sup> 14 11 1 *C C J(x) / x / x [ exp( x/ yeC x exp( C x )]* <sup>∞</sup> − − = −+ − − ⋅ − (22)

2 4 4 2 4 2

*C C C C CC*

− − − − − −

*E* θand '

ΦΦΦ can be got through equations from (15) to (17). Using the values

2 1 1

ij *E* θ

(16)

(17)

(18)

(23)

 θ

7. ij ij ij

The functions, ij

*' , ,* φ

In equations (15), (16) and (17),

 and ' ij *E* θ

> θ

θ

' 2

*E* θ

θ

fitted as the following function:

4 4

*' , ,* φ

finally, ij ij ij

*' , ,* φ

equations from (3) to (6).

compositions at equilibrium.

of ij ij ij

given the following form (Pitzer, 1973).

systems, and even geological fluids (Felmy & Weare, 1986; Kim & Frederich, 1988a, 1988b; Fang et al., 1993; Song, 1998; Song & Yao, 2001, 2003; Yang, 1988, 1989, 1992, 2005).

By the way, additional work has centered on developing variable-temperature models, which will increase the applicability to a number of diverse geochemical systems. The primary focus has been to broaden the models by generating parameters at higher or lower temperatures (Pabalan & Pitzer, 1987; Spencer et al., 1990; Greenberg & Moller, 1989).

#### **5.2 Model parameterization and solubility predictions**

As to the borate solution, the crystallized behavior of borate salts is very complex. The coexisted polyanion species of borate in the liquid phase is difference with the differences of boron concentration, pH value, solvent, and the positively charged ions. The ion of B4O72- is the general statistical express for various possible existed borates. Therefore, the structural formulas of Na2B4O7·10H2O and K2B4O7·4H2O in the solid phases of the quaternary system (NaCl - KCl – Na2B4O7 - K2B4O7 - H2O) are Na2[B4O5(OH)4]·8H2O and K2[B4O5(OH)4]·2H2O, respectively. Borate in the liquid phase corresponding to the equilibrium solid phase maybe coexists as B4O5(OH)42-, B3O3(OH)4 -, B(OH)4-, and son on due to the reactions of polymerization or depolymerization of boron anion.

Therefore, in this part of predictive solubility of the quaternary system (NaCl - KCl – Na2B4O7 - K2B4O7 - H2O), the predictive solubilities of this system were calculated on the basis of two assumptions: Model I: borate in the liquid phase exists all in statistical form of B4O7 2- i.e. B4O5(OH)42-; Model II: borate in the liquid phase exists as various boron species of B4O5(OH)42-, B3O3(OH)4-, B(OH)4- .

The necessary model parameters for the activity coefficients of electrolytes in the system at 298.15 K were fit from obtained osmotic coefficients and the sub-ternary subsystems by the multiple and unary linear regression methods.

#### **5.2.1 Model I for the solubility prediction**

Model I: Suppose that borate in solution exists as in the statistical expression form of B4O7 2 i.e. B4O5(OH)42-, and the dissolved equilibria in the system could be following:

$$\begin{aligned} \mathrm{Na\_2B\_4O\_7} \cdot 10 \mathrm{H\_2O} &= 2 \mathrm{Na^+} + \mathrm{B\_4O\_5(OH)\_4^{2-}} + 8 \mathrm{H\_2O} \\ \mathrm{K\_2B\_4O\_7} \cdot 4 \mathrm{H\_2O} &= 2 \mathrm{K^+} + \mathrm{B\_4O\_5(OH)\_4^{2-}} + 2 \mathrm{H\_2O} \\ \mathrm{NaCl} &= \mathrm{Na^+} + \mathrm{Cl^-} \\ \mathrm{KCl} &= \mathrm{K^+} + \mathrm{Cl^-} \end{aligned}$$

So, the dissolved equilibrium constants can be expressed as:

$$K\_{\rm N10} = (m\_{\rm Na^{+}} \cdot \mathcal{Y}\_{\rm Na^{+}})^2 \cdot (m\_{\rm B4} \cdot \mathcal{Y}\_{\rm B4}) \cdot a\_w^{\rm 8} \tag{24}$$

$$K\_{K4} = (m\_{K^+} \cdot \mathcal{Y}\_{K^+})^2 \cdot (m\_{B4} \cdot \mathcal{Y}\_{B4}) \cdot a\_w^{-2} \tag{25}$$

$$K\_{\rm NaCl} = (m\_{\rm Na^{+}} \cdot \mathcal{Y}\_{\rm Na^{+}}) \cdot (m\_{\rm Cl^{-}} \cdot \mathcal{Y}\_{\rm Cl^{-}}) \tag{26}$$

Stable and Metastable Phase Equilibria in the Salt-Water Systems 421

 *KCl K K Cl Cl K (m ) (m )* = ⋅⋅ ⋅ ++ − − γ

expresses the equilibrium constant of the polymerized species reaction of B4O5(OH)4

molalities of B3 and B are in equal. In the meantime, we suppose that two-ion and triplicateion interaction of different boron species would be weak, and the mixture ions parameters

Similar as in model I, then, the equilibrium constant *K* existed solid phase is calculated with *μ*0/*RT*, and also shown in Table 6, where another four possible borate salts of NaB3O3(OH)4, NaB(OH)4, KB3O3(OH)4, KB(OH)4 were also listed. The single salt parameters, binary ion interaction parameters, triplet mixture parameters and more parameters of θCl,B3O3(OH)4, θCl,B(OH)4,ΨCl,B3O3(OH)4,Na, and ΨCl,B(OH)4,Na were considered, and shown in Table 8. According to the equilibria constants and the Pitzer ion-interaction parameters, the solubilities of the quaternary system at 298.15 K have been calculated though the Newton's Iteration Method to solve the non-linearity simultaneous equations system, and shown in Table 10. In fact, this theoretic calculation for the reciprocal quaternary system is equivalence of the calculated solubilities for the six-component system (Na – K – Cl - B4O5(OH)4 - B3O3(OH)4 - B(OH)4 – H2O). It is worthy saying that although the concentrations of Na+, K+, Cl-, B4O5(OH)42-, B3O3(OH)4-, B(OH)4- in molalities could be got (Table 10), the concentrations including B4O5(OH)42-, B3O3(OH)4-, B(OH)4 should be all inverted into the concentration of B4O72- when the Jänecke index of B4O72-

Species *μ*0/*RT* Refs Species *μ*0/*RT* Refs

B 1984 3O3(OH)4- -963.77 KCl -164.84

Table 6. *μ*0/*RT* of species in the system (NaCl- KCl – Na2B4O7 - K2B4O7 - H2O) at 298.15 K

On the basis of the calculated solubilities, a comparison diagram among model I, model II, experimental values for the reciprocal quaternary system at 298.15 K are shown in

Harvie et al., 1984

1986

Na+ -105.651 Na2B4O5(OH)4·8H2O -2224.16

From this reaction of B4, B3 and B, i.e. B4O5(OH)42- + 2H2O = B3O3(OH)4-

Where, B4, B3 and B to instead of B4O5(OH)42-, B3O3(OH)4-

B3O3(OH)4

calculation.

Figure 9.

H2O -95.6635

K+ -113.957

Cl- -52.955

B4O5(OH)42- -1239.10 Felmy & Weare,


. And the electric charge balance exists as:

of different boron species should be ignored.

 γ

<sup>2</sup> 45 4 33 4 <sup>4</sup>

B(OH)4- -465.20

NaCl -154.99 Harvie et al.,

K2B4O5(OH)4·2H2O -1663.47

2 *m mm m m m Na K Cl B O (OH ) B O (OH ) B(OH )* ++ − += + + + <sup>−</sup> − − (34)

(33)

for short; *KB*4*B*3*<sup>B</sup>*

+ B(OH)4-

Felmy & Weare, 1986 2-,

, the

, and B(OH)4-

$$\mathbf{K}\_{\rm KCl} = (\mathbf{m}\_{\rm K^+} \cdot \boldsymbol{\chi}\_{\rm K^+}) \cdot (\mathbf{m}\_{\rm Cl^-} \cdot \boldsymbol{\chi}\_{\rm Cl^-}) \tag{27}$$

And the electric charge balance exists as:

$$m\_{Na^{+}} + m\_{K^{+}} = m\_{Cl^{-}} + 2m\_{B\_{4}O\_{5}(OH)\_{4}^{2-}} \tag{28}$$

Where, *K*, *r*, *m*, and *a*w express equilibrium constant, activity coefficient, and water activity, and N10, K4 instead of the minerals of Na2B4O7·10H2O, K2B4O7·4H2O (the same in the following), respectively. Then, the equilibria constants *K* are calculated with *μ*0/*RT* and shown in Table 6.

The single salt parameters *β*(0), *β*(1), *C*(φ) of NaCl, KCl, Na2[B4O5(OH)4], and K2[B4O5(OH)4], two-ion interaction Pitzer parameters of θNa, K, θCl, B4O5(OH)4 and the triplicate-ion Pitzer parameters of *Ψ*Cl, B4O5(OH)4, Na, *Ψ*Cl, B4O5(OH)4, K, *Ψ*Na, K, Cl, *Ψ*Na, K, B4O5(OH)4 in the reciprocal quaternary system at 298.15 K were chosen from Harvie et al. (1984), Felmy & Weare (1986), Kim & Frederick (1988), and Deng (2001) and summarized in Tables 7 and 8.

According to the equilibria constants and the Pitzer ion-interaction parameters, the solubilities of the quaternary system at 298.15 K have been calculated though the Newton's Iteration Method to solve the non-linearity simultaneous equations system, and shown in Table 9.

#### **5.2.2 Model II for the solubility prediction**

Model II: Suppose that borate in solution exists as in various boron species of B4O5(OH)4 2-, B3O3(OH)4-, B(OH)4- to further describe the behaviors of the polymerization and depolymerization of borate anion in solution, and the dissolved equilibria in the system could be following:

$$\mathrm{Na\_2B\_4O\_7} \cdot 10\mathrm{H\_2O} = 2\mathrm{Na^+} + \mathrm{B\_4O\_5(OH)\_4^{2-}} + 8\mathrm{H\_2O}$$

$$\mathrm{K\_2B\_4O\_7} \cdot 4\mathrm{H\_2O} = 2\mathrm{K^+} + \mathrm{B\_4O\_5(OH)\_4^{2-}} + 2\mathrm{H\_2O}$$

$$\mathrm{B\_4O\_5(OH)\_4^{2-}} + 2\mathrm{H\_2O} = \mathrm{B\_3O\_3(OH)\_4^{-}} + \mathrm{B(OH)\_4^{-}}$$

$$\mathrm{NaCl} = \mathrm{Na^+} + \mathrm{Cl^-}$$

$$\mathrm{KCl} = \mathrm{K^+} + \mathrm{Cl^-}$$

So, the dissolved equilibrium constants can be expressed as:

$$K\_{N10} \quad = (m\_{\rm Na^{+}} \cdot \mathcal{Y}\_{\rm Na^{+}})^2 \cdot (m\_{\rm B4} \cdot \mathcal{Y}\_{\rm B4}) \cdot a\_w^{-8} \tag{29}$$

$$K\_{K4} \quad = (m\_{K^+} \cdot \mathcal{Y}\_{K^+})^2 \cdot (m\_{B4} \cdot \mathcal{Y}\_{B4}) \cdot a\_w^{-2} \tag{30}$$

$$K\_{B4B3B} = \frac{(m\_{B3} \cdot \mathcal{Y}\_{B3}) \cdot (m\_B \cdot \mathcal{Y}\_B)}{(m\_{B4} \cdot \mathcal{Y}\_{B4}) \cdot a\_w^{-2}} \tag{31}$$

$$K\_{\rm NaCl} = (m\_{\rm Na^{+}} \cdot \mathcal{Y}\_{\rm Na^{+}}) \cdot (m\_{\rm CT^{-}} \cdot \mathcal{Y}\_{\rm CT^{-}}) \tag{32}$$

$$K\_{\rm KCl} \quad = (m\_{\rm K^+} \cdot \mathcal{Y}\_{\rm K^+}) \cdot (m\_{\rm Cl^-} \cdot \mathcal{Y}\_{\rm Cl^-}) \tag{33}$$

Where, B4, B3 and B to instead of B4O5(OH)42-, B3O3(OH)4 - , and B(OH)4 - for short; *KB*4*B*3*<sup>B</sup>* expresses the equilibrium constant of the polymerized species reaction of B4O5(OH)4 2-, B3O3(OH)4 -, B(OH)4 - .

And the electric charge balance exists as:

420 Advances in Crystallization Processes

 *KCl K K Cl Cl K (m ) (m )* = ⋅⋅ ⋅ ++ − − γ

Where, *K*, *r*, *m*, and *a*w express equilibrium constant, activity coefficient, and water activity, and N10, K4 instead of the minerals of Na2B4O7·10H2O, K2B4O7·4H2O (the same in the following), respectively. Then, the equilibria constants *K* are calculated with *μ*0/*RT* and

The single salt parameters *β*(0), *β*(1), *C*(φ) of NaCl, KCl, Na2[B4O5(OH)4], and K2[B4O5(OH)4], two-ion interaction Pitzer parameters of θNa, K, θCl, B4O5(OH)4 and the triplicate-ion Pitzer parameters of *Ψ*Cl, B4O5(OH)4, Na, *Ψ*Cl, B4O5(OH)4, K, *Ψ*Na, K, Cl, *Ψ*Na, K, B4O5(OH)4 in the reciprocal quaternary system at 298.15 K were chosen from Harvie et al. (1984), Felmy & Weare (1986),

According to the equilibria constants and the Pitzer ion-interaction parameters, the solubilities of the quaternary system at 298.15 K have been calculated though the Newton's Iteration Method to solve the non-linearity simultaneous equations system, and shown in

Model II: Suppose that borate in solution exists as in various boron species of B4O5(OH)42-, B3O3(OH)4-, B(OH)4- to further describe the behaviors of the polymerization and depolymerization of borate anion in solution, and the dissolved equilibria in the system

> 2 8 *K (m ) (m ) a <sup>N</sup>*<sup>10</sup> *Na Na BB w* 4 4 = ⋅ ⋅ ⋅⋅ + + γ

 2 2 *K (m ) (m ) a <sup>K</sup>*<sup>4</sup> *K K BB w* 4 4 = ⋅ ⋅ ⋅⋅ + + γ

> 3 3 4 3 2 4 4 *B B BB*

*(m ) (m ) <sup>K</sup> (m ) a* γ

*NaCl Na Na Cl Cl K (m ) (m )* = ⋅⋅⋅ + + −− γ

*BB w*

γ

 γ

 γ

> γ

> > γ

⋅ ⋅⋅ <sup>=</sup> ⋅ ⋅ (31)

(29)

(30)

(32)

Kim & Frederick (1988), and Deng (2001) and summarized in Tables 7 and 8.

And the electric charge balance exists as:

**5.2.2 Model II for the solubility prediction** 

NaCl = Na+ + Cl-

KCl = K+ + Cl-

Na2B4O7·10H2O = 2Na+ + B4O5(OH)42- + 8H2O

 K2B4O7·4H2O = 2K+ + B4O5(OH)42- + 2H2O B4O5(OH)42- + 2H2O = B3O3(OH)4- + B(OH)4-

So, the dissolved equilibrium constants can be expressed as:

*BBB*

shown in Table 6.

Table 9.

could be following:

 γ

<sup>2</sup> 45 4 2 *Na K Cl B O (OH ) m mm m* ++ − += + <sup>−</sup> (28)

(27)

$$m\_{Na^{+}} + m\_{K^{+}} = m\_{Cl^{-}} + 2m\_{B\_{4}O\_{5}(OH)\_{4}^{2-}} + m\_{B\_{3}O\_{3}(OH)\_{4}^{-}} + m\_{B(OH)\_{4}^{-}} \tag{34}$$

From this reaction of B4, B3 and B, i.e. B4O5(OH)4 2- + 2H2O = B3O3(OH)4 - + B(OH)4- , the molalities of B3 and B are in equal. In the meantime, we suppose that two-ion and triplicateion interaction of different boron species would be weak, and the mixture ions parameters of different boron species should be ignored.

Similar as in model I, then, the equilibrium constant *K* existed solid phase is calculated with *μ*0/*RT*, and also shown in Table 6, where another four possible borate salts of NaB3O3(OH)4, NaB(OH)4, KB3O3(OH)4, KB(OH)4 were also listed. The single salt parameters, binary ion interaction parameters, triplet mixture parameters and more parameters of θCl,B3O3(OH)4, θCl,B(OH)4,ΨCl,B3O3(OH)4,Na, and ΨCl,B(OH)4,Na were considered, and shown in Table 8. According to the equilibria constants and the Pitzer ion-interaction parameters, the solubilities of the quaternary system at 298.15 K have been calculated though the Newton's Iteration Method to solve the non-linearity simultaneous equations system, and shown in Table 10. In fact, this theoretic calculation for the reciprocal quaternary system is equivalence of the calculated solubilities for the six-component system (Na – K – Cl - B4O5(OH)4 - B3O3(OH)4 - B(OH)4 – H2O). It is worthy saying that although the concentrations of Na+, K+, Cl-, B4O5(OH)42-, B3O3(OH)4-, B(OH)4- in molalities could be got (Table 10), the concentrations including B4O5(OH)42-, B3O3(OH)4-, B(OH)4 should be all inverted into the concentration of B4O72- when the Jänecke index of B4O72 calculation.


Table 6. *μ*0/*RT* of species in the system (NaCl- KCl – Na2B4O7 - K2B4O7 - H2O) at 298.15 K

On the basis of the calculated solubilities, a comparison diagram among model I, model II, experimental values for the reciprocal quaternary system at 298.15 K are shown in Figure 9.

Stable and Metastable Phase Equilibria in the Salt-Water Systems 423

1 0.2748 1.2843 0.00 0.7796 17.62 100.00 N10+K4 2 0.2768 1.3032 0.2000 0.6900 17.52 87.34 N10+K4 3 0.2804 1.3279 0.3500 0.6292 17.43 78.24 N10+K4 4 0.2863 1.3619 0.5000 0.5741 17.37 69.66 N10+K4 5 0.3014 1.4496 0.7800 0.4855 17.21 55.45 N10+K4 6 0.3177 1.5386 1.0000 0.4282 17.11 46.13 N10+K4 7 0.3603 1.7573 1.4400 0.3388 17.02 32.00 N10+K4 8 0.4297 2.0854 2.0000 0.2575 17.08 20.48 N10+K4 9 0.5028 2.4030 2.5000 0.2029 17.30 13.97 N10+K4 10 0.5858 2.7319 3.0000 0.1589 17.66 9.58 N10+K4 11 0.6774 3.0673 3.5000 0.1224 18.09 6.54 N10+K4 12 0.8903 3.7442 4.5000 0.06725 19.21 2.90 N10+K4 13,A1 1.0298 4.1687 5.1107 0.04392 19.81 1.69 N10+K4+KCl 14 0.00 4.8834 4.7149 0.08423 0.00 3.45 K4+KCl 15 0.1500 4.7688 4.7699 0.07444 3.05 3.03 K4+KCl 16 0.3000 4.6592 4.8262 0.06649 6.05 2.68 K4+KCl 17 0.4500 4.5534 4.8835 0.05996 8.99 2.40 K4+KCl 18 0.6000 4.4509 4.9418 0.05457 11.88 2.16 K4+KCl 19 0.7500 4.3511 5.0009 0.05009 14.70 1.96 K4+KCl 20 0.9000 4.2536 5.0609 0.04636 17.46 1.80 K4+KCl 21 4.8000 2.2523 7.0507 8.13E-4 68.06 0.023 N10+KCl 22 4.3000 2.4641 6.7621 9.67E-4 63.57 0.029 N10+KCl 23 3.6000 2.7821 6.3794 0.00139 56.41 0.044 N10+KCl 24 3.2000 2.9750 6.1713 0.00185 51.82 0.060 N10+KCl 25 2.8000 3.1758 5.9705 0.00263 46.86 0.088 N10+KCl 26 2.4000 3.3843 5.7762 0.00405 41.49 0.14 N10+KCl 27 2.0000 3.6007 5.5869 0.00686 35.71 0.25 N10+KCl 28 1.6000 3.8255 5.3992 0.01316 29.49 0.49 N10+KCl 29 5.2183 1.9000 7.1163 9.75E-4 73.31 0.027 N10+NaCl 30 5.4046 1.5000 6.9014 0.00159 78.28 0.046 N10+NaCl 31 5.6432 1.0000 6.6369 0.00313 84.95 0.094 N10+NaCl 32 5.8894 0.5000 6.3762 0.0066 92.17 0.21 N10+NaCl 33 6.1479 0.00 6.1178 0.01504 100.00 0.49 N10+NaCl 34,B1 5.1148 2.1256 7.2389 7.53E-4 70.64 0.021 N10+NaCl+K

35 5.1147 2.1259 7.2394 6.00E-4 70.64 0.017 NaCl+KCl 36 5.1145 2.1264 7.2401 4.00E-4 70.63 0.011 NaCl+KCl 37 5.1143 2.1269 7.2408 2.00E-4 70.63 0.0055 NaCl+KCl 38 5.1142 2.1273 7.2415 0.00 70.62 0.00 NaCl+KCl

Table 9. Calculated solubility data of the system (NaCl- KCl – Na2B4O7 - K2B4O7 - H2O) at

298.15 K on the basis of Model I. \* N10, Na2B4O7·10H2O; K4, K2B4O7·4H2O.

Jänecke index, *J*/(mol/100mol dry salts)

B4O72- *J*(2Na+) *J*(B4O72-)

Equilibrium solid phases\*

Cl

Composition liquid phase molality, /(mol/kgH2O)

Na+ K+ Cl-

No.


Table 7. Single-salt Pitzer parameters in the system (NaCl- KCl – Na2B4O7 - K2B4O7 - H2O) at 298.15 K


Table 8. Mixing ion-interaction Pitzer parameters in the system (NaCl - KCl – Na2B4O7 - K2B4O7 - H2O) at 298.15 K

Though the theoretical calculation on the basis of model II, it was found that the boron species are mainly existed B3O3(OH)4 - and B(OH)4 - while the concentration of B4O5(OH)4 2- is very low when the total concentration of boron is low in weak solution. This result demonstrated that the polymerization or depolymerization behaviors of borate are complex.


Na+ Cl- 0.07722 0.25183 0.00106 Kim & Frederick, 1988

Na Felmy & Weare, 1986 + B3O3(OH)4-

K+ B3O3(OH)4 Felmy & Weare, 1986 -

Table 7. Single-salt Pitzer parameters in the system (NaCl- KCl – Na2B4O7 - K2B4O7 - H2O) at

K+ Cl- 0.04835 0.2122 -0.00084 Harvie et al., 1984



θNa+, K+ -0.012 Harvie et al., 1984

θCl Felmy & Weare, 1986 -

ΨCl-, B3O3(OH)4-, Na+ -0.024 Felmy & Weare, 1986

*( )*

*MX C* φ

Refs

Cation Anion *( )* <sup>0</sup>

298.15 K

θCl-

θCl-

*MX* β

Na+ B4O5(OH)42- -0.11 -0.40 0.0

Na+ B(OH)4- -0.0427 0.089 0.0114

K+ B4O5(OH)42- -0.022 0.0 0.0

K+ B(OH)4- 0.035 0.14 0.0

, B4O5(OH)42- 0.074

, B3O3(OH)4- 0.12

, B(OH)4- -0.065

ΨCl-, B4O5(OH)42-, Na+ 0.025

ΨCl-, B(OH)4-, Na+ -0.0073

Parameters Values Refs

θB4O5(OH)42-, B3O3(OH)4- — θB4O5(OH)42-, B(OH)4- — θB3O3(OH)4-, B(OH)4- — —

ΨB4O5(OH)42-, B3O3(OH)4-, Na+ — — ΨB4O5(OH)42-, B(OH)4-, Na+ — — ΨB3O3(OH)4-, B(OH)4-, Na+ — —

ΨCl-, B3O3(OH)4-, K+ — — ΨCl-, B(OH)4-, K+ — — ΨB4O5(OH)42-, B3O3(OH)4-, K+ — — ΨB4O5(OH)42-, B(OH)4-, K+ — — ΨB3O3(OH)4-, B(OH)4-, K+ — —

ΨNa+, K+, B3O3(OH)4- — — ΨNa+, K+, B(OH)4- — —

K2B4O7 - H2O) at 298.15 K

species are mainly existed B3O3(OH)4

ΨCl-, B4O5(OH)42-, K+ 0.0185245 Deng, 2004

ΨNa+, K+, Cl- -0.0018 Harvie et al., 1984 ΨNa+, K+, B4O5(OH)42- 0.289823 Deng, 2004

Table 8. Mixing ion-interaction Pitzer parameters in the system (NaCl - KCl – Na2B4O7 -


Though the theoretical calculation on the basis of model II, it was found that the boron

very low when the total concentration of boron is low in weak solution. This result demonstrated that the polymerization or depolymerization behaviors of borate are complex.


2- is

 *MX C*φ *( )* 1 *MX* β


Table 9. Calculated solubility data of the system (NaCl- KCl – Na2B4O7 - K2B4O7 - H2O) at 298.15 K on the basis of Model I. \* N10, Na2B4O7·10H2O; K4, K2B4O7·4H2O.

Stable and Metastable Phase Equilibria in the Salt-Water Systems 425

In Figure 9, compared with Models I and II, the calculated values in the boundary points and the cosaturated point of (Na2B4O7·10H2O + KCl + NaCl) based on model II were in good agreement with the experimental data. However, in the cosaturated point of (Na2B4O7·10H2O + K2B4O7·4H2O + KCl), a large difference on the solubility curve still existed. Reversely, the predictive result based on model II closed to the experimental curve. There were two possible reasons: one is that the structure of borate in solution is very complex, an the Pitzer's parameters of borate salts is scarce; the other one is the high saturation degree of borate, the difference between the experimental equilibrium constant

0 10 20 30 40 50 60 70 80 90 100

*J*(2Na+ )

Fig. 9. Comparison of the experimental and calculated phase diagram of the quaternary system (NaCl - KCl – Na2B4O7 - K2B4O7 - H2O) at 298.15 K. -●-, Calculated based on Model

Financial support from the State Key Program of NNSFC (Grant.20836009), the NNSFC (Grant. 40773045), the "A Hundred Talents Program" of the Chinese Academy of Sciences (Grant. 0560051057), the Specialized Research Fund for the Doctoral Program of Chinese Higher Education (Grant 20101208110003), The Key Pillar Program in the Tianjin Municipal Science and Technology (11ZCKFGX2800) and Senior Professor Program of Tianjin for TUST (20100405) is acknowledged. Author also hopes to thank all members in my research group and my Ph.D. students Y.H. Liu, S.Q. Wang, Y.F. Guo, X.P. YU, J. Gao, L.Z. Meng, DC Li, Y.

Analytical Laboratory of Institute of Salt Lakes at CAS. (1988). *The analytical methods of brines and salts*, 2nd ed., Chin. Sci. Press, ISBN 7-03-000637-2, pp. 35-41 & 64-66, Beijing

E F B2 A2

Borax

B1

 Model I Model II -- O -- Experimental B4 O7

2NaCl

O7 Na2

and the theoretic calculated equilibrium constant was large enough.

*J*(B

O4 72-) K2 B4

2KCl

I; -▲-, Calculated based on Model II; -○-, Experimental.

**6. Acknowledgments** 

**7. References** 

A1

Wu, and D.M. Lai for their active contributions on our scientific projects.


Table 10. Calculated solubility data of the system (NaCl- KCl – Na2B4O7 - K2B4O7 - H2O) at 298.15 K on the basis of Model II. \* B4, B3, B express for B4O5(OH)42-, B3O3(OH)4 -, B(OH)4-; N10, Na2B4O7·10H2O; K4, K2B4O7·4H2O.

1 0.3362 1.4923 0.00 0.7084 0.2058 0.2058 18.38 100.00 N10+K4 2 0.3394 1.5158 0.2000 0.6362 0.1914 0.1914 18.28 89.22 N10+K4 3 0.3431 1.5428 0.3500 0.5868 0.1812 0.1812 18.19 81.44 N10+K4 4 0.3485 1.5771 0.5000 0.5413 0.1715 0.1715 18.10 74.03 N10+K4 5 0.3628 1.6608 0.7800 0.4667 0.1551 0.1551 17.93 61.45 N10+K4 6 0.3777 1.7431 1.0000 0.4165 0.1439 0.1439 17.81 52.85 N10+K4 7 0.4168 1.9432 1.4400 0.3347 0.1253 0.1253 17.66 38.98 N10+K4 8 0.4818 2.2464 2.0000 0.2562 0.1079 0.1079 17.66 26.69 N10+K4 9 0.5519 2.5451 2.5000 0.2018 0.0967 0.0967 17.82 19.28 N10+K4 10 0.6326 2.8591 3.0000 0.1574 0.0884 0.0884 18.12 14.08 N10+K4 11 0.7234 3.1822 3.5000 0.1207 0.0821 0.0821 18.52 10.38 N10+K4 12 0.9355 3.8426 4.5000 0.0654 0.0736 0.0736 19.58 5.82 N10+K4 13,A2 1.0770 4.2295 5.0794 0.04305 0.0705 0.0705 20.30 4.28 N10+K4+KCl 14 0.00 4.9588 4.6844 0.08694 0.0502 0.0502 0.00 5.53 K4+KCl 15 0.1500 4.8452 4.7369 0.07661 0.0525 0.0525 3.00 5.17 K4+KCl 16 0.3000 4.7368 4.7905 0.06822 0.0549 0.0549 5.96 4.89 K4+KCl 17 0.4500 4.6326 4.8449 0.06133 0.0576 0.0576 8.85 4.68 K4+KCl 18 0.6000 4.5320 4.8999 0.05563 0.0604 0.0604 11.69 4.52 K4+KCl 19 0.7500 4.4342 4.9558 0.05089 0.0633 0.0633 14.47 4.41 K4+KCl 20 0.9000 4.3390 5.0122 0.04692 0.0665 0.0665 17.18 4.33 K4+KCl 21 4.8000 2.2665 7.0005 8.237E-4 0.0322 0.0322 67.93 0.93 N10+KCl 22 4.3000 2.4815 6.7150 9.779E-4 0.0323 0.0323 63.41 0.98 N10+KCl 23 3.6000 2.8052 6.3356 0.00141 0.0334 0.0334 56.20 1.09 N10+KCl 24 3.2000 3.0021 6.1289 0.00188 0.0347 0.0347 51.60 1.18 N10+KCl 25 2.8000 3.2078 5.9289 0.00268 0.0368 0.0368 46.61 1.31 N10+KCl 26 2.4000 3.4226 5.7346 0.00415 0.0398 0.0398 41.22 1.51 N10+KCl 27 2.0000 3.6474 5.5442 0.0071 0.0445 0.0445 35.41 1.83 N10+KCl 28 1.6000 3.8845 5.3532 0.0138 0.0519 0.0519 29.17 2.40 N10+KCl 29 5.2316 1.9000 7.0663 9.963E-4 0.0316 0.0316 73.36 0.91 N10+NaCl 30 5.4169 1.5000 6.8530 0.00163 0.0303 0.0303 78.31 0.92 N10+NaCl 31 5.6544 1.0000 6.5902 0.00321 0.0289 0.0289 84.97 0.96 N10+NaCl 32 5.8998 0.5000 6.3310 0.00679 0.0276 0.0276 92.19 1.07 N10+NaCl 33 6.1580 0.00 6.0739 0.0155 0.0266 0.0266 100.00 1.37 N10+NaCl 34,B2 5.1668 2.1309 7.2304 7.463E-4 0.0329 0.0329 70.80 0.92 N10+KCl+NaCl 35 5.1612 2.1307 7.2318 6.00E-4 0.0294 0.0294 70.78 0.82 NaCl+KCl 36 5.1523 2.1303 7.2339 4.00E-4 0.0239 0.0239 70.75 0.67 NaCl+KCl 37 5.1410 2.1296 7.2364 2.00E-4 0.0168 0.0168 70.71 0.47 NaCl+KCl 38 5.1142 2.1273 7.2415 0.00 0.00 0.00 70.62 0.00 NaCl+KCl Table 10. Calculated solubility data of the system (NaCl- KCl – Na2B4O7 - K2B4O7 - H2O) at 298.15 K on the basis of Model II. \* B4, B3, B express for B4O5(OH)42-, B3O3(OH)4-, B(OH)4-;

B4 B3 B J(2Na+) J(B4O72-)

Jänecke index, *J* **/(mol/100mol dry-salt)** 

Equilibrium solid phases

Composition liquid phase molality, /(mol/kgH2O)\*

Na+ K+ Cl-

N10, Na2B4O7·10H2O; K4, K2B4O7·4H2O.

No.

In Figure 9, compared with Models I and II, the calculated values in the boundary points and the cosaturated point of (Na2B4O7·10H2O + KCl + NaCl) based on model II were in good agreement with the experimental data. However, in the cosaturated point of (Na2B4O7·10H2O + K2B4O7·4H2O + KCl), a large difference on the solubility curve still existed. Reversely, the predictive result based on model II closed to the experimental curve. There were two possible reasons: one is that the structure of borate in solution is very complex, an the Pitzer's parameters of borate salts is scarce; the other one is the high saturation degree of borate, the difference between the experimental equilibrium constant and the theoretic calculated equilibrium constant was large enough.

Fig. 9. Comparison of the experimental and calculated phase diagram of the quaternary system (NaCl - KCl – Na2B4O7 - K2B4O7 - H2O) at 298.15 K. -●-, Calculated based on Model I; -▲-, Calculated based on Model II; -○-, Experimental.

#### **6. Acknowledgments**

Financial support from the State Key Program of NNSFC (Grant.20836009), the NNSFC (Grant. 40773045), the "A Hundred Talents Program" of the Chinese Academy of Sciences (Grant. 0560051057), the Specialized Research Fund for the Doctoral Program of Chinese Higher Education (Grant 20101208110003), The Key Pillar Program in the Tianjin Municipal Science and Technology (11ZCKFGX2800) and Senior Professor Program of Tianjin for TUST (20100405) is acknowledged. Author also hopes to thank all members in my research group and my Ph.D. students Y.H. Liu, S.Q. Wang, Y.F. Guo, X.P. YU, J. Gao, L.Z. Meng, DC Li, Y. Wu, and D.M. Lai for their active contributions on our scientific projects.

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**17** 

**"Salt Weathering" Distress** 

Zanqun Liu1,2.3, Geert De Schutter2, Dehua Deng1.3 and Zhiwu Yu1,3 *1School of Civil Engineering, Central South University, Changsha, Hunan,* 

*2Magnel Laboratory for Concrete Research, Department of Structural Engineering,* 

Salt weathering, also called salt crystallization or physical salt attack, is defined as the basic degradation mechanism that a porous material, such as stone and masonry, undergoes at and near the Earth's surface [1]. The parts of porous materials in contact with relatively dry air near the Earth's surface will be severely deteriorated but the parts buried in salts

Generally, the idea of sulfate attack on concrete means that a complex physiochemical process including several harmful productions formation through chemical reaction, such as ettringite and gypsum, following the crystal growth of these productions in cracks or pores resulting in concrete damage. However, another concept was given more and more attention that "salt weathering/physical salt attack" on concrete partially exposed to environment specially containing Na2SO4 or MgSO4. ACI (American Concrete Institute) created a new subcommittee, ACI 201-E (Salt Weathering/Physical Salt Attack) in 2009. In 2011, an ballot was performed to discuss if it is necessary to separate the "physical salt attack" from chapter 6 "sulfate attack" as chapter 8 for ACI 201.2R. There were also more and more reports discussing this topic [2-9]. It seems that this topic will be high interest and

Certainly, concrete is also a kind of porous material. When partially exposed to an environment containing salts (especially sodium sulfate), such as in the case of a foundation, dam, column, flatwork and tunnel, a large amount of efflorescence will appear on the surface of the concrete accompanied with a similar scaling manner as salt weathering distress on masonry, showing a freezing-and-thawing-like deterioration on the surface of concrete [2] (Fig. 1). Therefore, concrete technologists logically and involuntarily define this

Apparently, it seems reasonable to attribute salt weathering to the decay of concrete partially exposed to sulfate environment. Concrete technologists subjectively accepted that

phenomenon as salt weathering distress on concrete or physical attack on concrete.

**1. Introduction** 

environment look sound.

relevance for the concrete community.

*3National Engineering Laboratory for High Speed Railway Construction,* 

**on Concrete by Sulfates?** 

*Ghent University, Ghent,* 

*Changsha, Hunan, 1,3P.R China 2Belgium* 

