Thermodynamic Modeling of Boron Species in the Ternary System Na2O-B2O3-H2O at 298.15 K

Fu, Qingtao; Chen, Lele; Song, Xiaohui; Li, Dan; Yang, Xiuxiu; Meng, Lingzong; Guo, Yafei; Deng, Tianlong

doi:https://doi.org/10.1155/2020/6687742

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Research Article | Open Access

Volume 2020 | Article ID 6687742 | https://doi.org/10.1155/2020/6687742

Thermodynamic Modeling of Boron Species in the Ternary System Na₂O-B₂O₃-H₂O at 298.15 K

Qingtao Fu,¹Lele Chen,²Xiaohui Song,¹Dan Li,¹Xiuxiu Yang,²Lingzong Meng,¹Yafei Guo,²and Tianlong Deng²

Academic Editor: Gang Feng

Received03 Nov 2020

Revised30 Nov 2020

Accepted07 Dec 2020

Published16 Dec 2020

Abstract

The solubilities; concentration of four boron species B(OH)₃, B(OH)₄⁻, B₃O₃(OH)₄⁻, and B₄O₅(OH)₄²⁻; and H⁺ (OH^–) in the system Na₂O–B₂O₃–H₂O were calculated with the Pitzer model. The boron species B₅O₆(OH)₄⁻ was not considered in the model calculation. The calculated solubility and pH data are in accordance with the experimental results. The Pitzer model can be used to describe the experimental values. The distributions of the four boron species in the solution were obtained. The mole fractions of the four boron species are mainly affected by the ratio of B₂O₃ and Na₂O in the solution. The dominant boron species in the solution saturated with different salts vary greatly. The results can supply a theoretical reference for sodium borate separation from brine.

1. Introduction

The Qaidam Basin in Qinghai is the gathering place of salt lake resources in China [1]. In addition to the rich mineral resources such as potassium, sodium, magnesium, and lithium, the salt lake brine also has a large amount of borates [2–5]. These resources have important application value for the development of industry, agriculture, and economic construction [6]. There are many boron species in the borate solution. The boron species have complex structure and are mainly affected by the boron concentration, pH, and coexisting ions in solution [7–9]. The phase equilibria of water-salt systems can supply theoretical basis for the development of salt lake brine resources [10]. Therefore, the research on the phase equilibrium of the brine systems containing boron is meaningful to the comprehensive utilization and development of boron resources from brines [11].

The system Na₂O-B₂O₃-H₂O is the simple subsystem of the complicated brine system. The phase equilibria in the system in a wide temperature range (273.15∼373.15 K) were reported [12–15]. The salts H₃BO₃, NaB₅O₆(OH)₄ · 3H₂O, Na₂B₄O₅(OH)₄ · 8H₂O, and NaB(OH)₄ · 2H₂O were found in the system at 298.15 K [12–14]. The different salt forms with different ratios of Na₂O and B₂O₃ in the solution. The change trends for concentration of total boron in the solution can be obtained with the phase equilibrium results, but the exact concentration of various boron species in the solution cannot be obtained with the phase equilibrium results.

The thermodynamic model is the effective method to calculate the concentration of boron species [16]. The MSE model was applied to calculate the solubilities in the system Na₂O-B₂O₃-H₂O at 303.15 K, 333.15 K, 348.15 K, and 367.15 K [17]. However, the concentrations of different boron species and H⁺ (OH^–) in the system were not presented [17]. The Pitzer model was widely applied in the solubility calculation in the brine systems [18, 19]. The Pitzer model in the system Na⁺-K⁺-Ca²⁺-Mg²⁺-H⁺-Cl^–-SO₄^2–-CO₂-B(OH)₄^–-H₂O was reported by Felmy and Weare [20]. The solubilities in the system Na₂O-B₂O₃-H₂O at 293.15 K were reported with the Pitzer model. The concentration of H⁺ (OH^–) in the system containing boric acid or borate cannot be neglected [21, 22]. However, H⁺ (OH^–) was not considered in the Pitzer model for the system. The concentrations of different boron species in the ternary system were not yet calculated [20]. The solubilities in the systems NaCl-NaBO₂-Na₂B₄O₇-H₂O and NaCl-Na₂SO₄-NaBO₂-H₂O at 298.15 K were calculated with Pitzer model in our previous work [21, 22]. The concentrations of various boron species and H⁺ in the mixed borate solution were also calculated. However, the Pitzer model containing H⁺ (OH^–) in the system Na₂O-B₂O₃-H₂O at 298.15 K, which is necessary for synthesis of sodium borate, is lacking. In this paper, the Pitzer model in the system Na₂O-B₂O₃-H₂O at 298.15 K was constructed, and the concentrations of various boron species and H⁺ (OH^–) in the system were also calculated with the model.

2. Model Approach

The Pitzer thermodynamic model was widely used to calculate solubility, osmotic coefficient ø, ionic activity coefficients γ, and other thermodynamic properties [23–26]. These main equations for thermodynamic property calculation are as follows:

In the above equations, A^Ø represent Debye–Hückel parameters. The subscripts M, c, c′, X, a, a′, and N express the different cations and anions, respectively. The symbols m_c and Z_c are the mass molality and the charge of cation c. λ_Nc,λ_Na, and ζ_Nca represent the interaction between the cations, anions, ionic species, and the neutral species. The other terms F, C, Z, B, and Φ in the above equations are defined in references [23–26]. In equation (5), α_w and M_W represent the water activity and molar mass of water (mol · kg⁻¹), and the sum contains all solute species.

The solubilities of brine systems can be calculated with the Pitzer parameters and solubility products of the equilibrium solid phases (K_sp). In this study, the solubilities of the system Na₂O-B₂O₃-H₂O at 298.15 K were calculated with the model based on the Pitzer model. The pH for the solution in the system Na₂O-B₂O₃-H₂O at 298.15 K is in the range from 2.50 to 14.46 [12,27], which shows the concentration of H⁺ or OH^–cannot be neglected in the solution. Therefore, the ions Na⁺, H⁺ (OH⁻), B(OH)₃, B(OH)₄⁻, B₃O₃(OH)₄⁻, and B₄O₅(OH)₄²⁻ in the solution were considered in the model calculation. The boron ion B₅O₆(OH)₄⁻ was not mentioned in the calculation because of the lack of parameters for B₅O₆(OH)₄⁻. The Pitzer parameters and /RT of different boron species in the system Na₂O-B₂O₃-H₂O at 298.15 K were obtained from Felmy and Weare [20]. The lacking Pitzer mixing ion-interaction parameters for boron ions are considered as zero.

Taking the solution saturated with H₃BO₃ in the ternary system for example, the dissolution equilibrium equations are as follows, and the relationship between boron ions can be expressed in equations (6)–(9).

The equilibrium constants for equations (6)–(9), which can be calculated with the standard chemical potentials by Felmy and Weare [20, 26], are shown in equations (10)–(13).

In equations (10)–(13), K₁, K₂, K₃, and K₄ represent the equilibrium constant equations for equations (6)–(9). The solution is saturated with sodium borate when B(OH)₃ is unsaturated, and the solution is alkaline, taking NaB₄O₅(OH)₄ · 8H₂O for example. The dissolution equilibrium equations are as follows. The relationship between boron species can be represented in equations (15)–(17).

The solubility product constant of NaB₄O₅(OH)₄ · 8H₂O (K₅) at a certain temperature and pressure is shown in equation (18). The equilibrium constants for equations (15)–(17) are shown in equations (19)–(21):

In equations (18)–(21), K₅, K₆, K₇, and K₈ represent the equilibrium constant equations for equations (18)–(21).

B₅O₆(OH)₄⁻ cannot be neglected in the solution in the system NaB₅O₆(OH)₄-H₂O when the concentration of NaB₅O₆(OH)₄ is not very low [7]. In the system Na₂O-B₂O₃-H₂O, the concentration of B₅O₆(OH)₄⁻ may be not neglected when the solution is saturated with NaB₅O₆(OH)₄·3H₂O. B₅O₆(OH)₄⁻ will convert into B(OH)₃ and B₃O₃(OH)₄⁻ when the concentration of NaB₅O₆(OH)₄ is low. Therefore, the dissolution equilibrium constant of B₅O₆(OH)₄⁻ (K₉) can be calculated with equation (23) because of the lacking parameters for B₅O₆(OH)₄⁻:

According to the Pitzer model, the activity and osmotic coefficients are parametric functions of β⁽⁰⁾, β⁽¹⁾, β⁽²⁾, C^Ø, θ_aa′, ψ_aa′c, λ_Nc,λ_Na, and ζ_Nca. β⁽⁰⁾, β⁽¹⁾, β⁽²⁾, and C^Ø are the parameters of a single salt; θ_aa′ represents the interaction between the two ions with the same sign, and ψ_aa′c represents the interactions among the three ions, in which the sign of the third ion is different from the first two ions. Combining the charge conservation and matter conservation equations, the compositions of different species at equilibrium and solubilities for the system Na₂O-B₂O₃-H₂O can be calculated with the above equations with the Pitzer parameters and standard chemical potentials (/RT) of different species.

3. Result and Discussion

The experimental and calculated solubilities of boundary point saturated with B(OH)₃ and invariant points for the ternary system Na₂O-B₂O₃-H₂O at 298.15 K are shown in Table 1. The comparisons of experimental and calculated phase diagrams in the system at 298.15 K are shown in Figure 1. The point A, which is saturated with B(OH)₃, is the boundary point in the system Na₂O-B₂O₃-H₂O. There are three invariant points (E₁, E₂, and E₃) and four solubility curves in the system. According to Table 1 and Figure 1, the calculated values are in agreement with the experimental values except point E₃. The results show that the Pitzer model with four boron species is reliable for calculating the solubilities in the system Na₂O-B₂O₃-H₂O. From A to E₂, (B₂O₃) increased as (Na₂O) increases and reaches the maximum data at point E₂. The point with minimum (B₂O₃) in Figure 1 is shown as point B. The solubility curve saturated with Na₂B₄O₅(OH)₄ · 8H₂O can be separated into two parts E₂B and BE₃. From E₂ to B, (B₂O₃) decreased sharply as (Na₂O) decreased. However, (B₂O₃) increased relatively gently as (Na₂O) increases from B to E₃. In the solubility curve saturated with NaB(OH)₄ · 2H₂O, (B₂O₃) decreased with (Na₂O) increasing.

The H⁺ or OH⁻ concentration cannot be neglected in the system Na₂O-B₂O₃-H₂O at 298.15 K. The calculated and experimental pH diagrams were plotted, as shown in Figure 2. The mass fractions (100) of Na₂O were used as the abscissa in Figure 2. The calculated pH data are in accordance with the experimental results except some points. The pH datum for point A is 4.27 from Валяшко [12] and 2.50 from Li [27] in Table 1. Considering the experimental error, the calculated data for point A (3.15) can be considered to be in agreement with the experimental data. The pH data increased with increasing (Na₂O). The changing trend varied in different curves saturated with different salts. With the calculated pH diagram, the invariant points in the system can be roughly judged.

The reasons about the difference between the calculated solubility and pH data with experimental results are complicated. On the one hand, only four boron species were considered in the solution of the system Na₂O-B₂O₃-H₂O in this model calculation. However, B₅O₆(OH)₄^– may exist in the solution in the system [7]. Moreover, the concentration of B₅O₆(OH)₄^– cannot be neglected as m (B) in the solution increases [7]. On the other hand, the lacking Pitzer parameters for boron ions cannot be considered as zero. Therefore, the difference between the experimental and calculated values is inevitable, and it was controlled to the minimum based on our current knowledge.

The mole fractions (x_i) for the four boron species in this ternary system calculated with the Pitzer model are shown in Figure 3. The mole fraction of the four boron species (x_i) can be calculated with equations (24)–(27), which is the same as our previous work [22]. In equations (24)–(27), B₄^2–, B₃^–, B^–,and B represent B₄O₅(OH)₄^2–, B₃O₃(OH)₄^–, B(OH)₄^–, and B(OH)₃.

(a)

(b)

(c)

(d)

The variation trends for the relationships between x_i and (Na₂O) are different in Figure 3. With the difference, the invariant points can be judged. From Figure 3(a), x(B₄O₅(OH)₄^2–) increased from point A to E₂ as (Na₂O) increased. In the curve E₂E₃ saturated with Na₂B₄O₅(OH)₄ · 8H₂O, x(B₄O₅(OH)₄^2–) firstly increased sharply as (Na₂O) decreased, and then decreased sharply as (Na₂O) increased. In the curve saturated with NaB(OH)₄ · 2H₂O in Figure 3(a), x(B₄O₅(OH)₄²⁻) decreased to nearly zero. In Figure 3(b), x(B₃O₃(OH)₄^–) increased from points A to E₂. From points E₂ to E₃, x(B₃O₃(OH)₄^–) decreased to nearly zero. x(B₃O₃(OH)₄^–) can be neglected in the curve saturated with NaB(OH)₄ · 2H₂O. x(B(OH)₄^–) in Figure 3(c) is nearly zero in the curves AE₁ and E₁E₂. x(B(OH)₄^–) increased from points E₂ to E₃ and reached the maximum data at E₃. In the curve saturated with NaB(OH)₄ · 2H₂O, x (B(OH)₄^–) increased sharply to the maximum and became stable with the mole fraction no less than 0.99. In Figure 3(d), x (B(OH)₃) decreased as (Na₂O) increased from point A to E₂. In the curve E₂E₃, x (B(OH)₃) decreased to nearly zero. x (B(OH)₃) can be neglected if (Na₂O) is more than 0.028. From Figure 3, the mole fraction of the four boron species is mainly affected by the ratio of B₂O₃ and Na₂O in the solution, which affects the equilibrium solid phase in the system Na₂O-B₂O₃-H₂O.

From Figure 3, the dominant boron species in solutions saturated with different salts vary greatly. The main boron species are B(OH)₃ and B₃O₃(OH)₄^– in the solution saturated with B(OH)₃ or NaB₅O₆(OH)₄·3H₂O. B(OH)₄^– is the dominant boron species in the solution saturated with NaB(OH)₄·2H₂O. The boron species are very different in the solution saturated with Na₂B₄O₅(OH)₄·3H₂O. When (Na₂O) is less than 0.028, the four boron species exist in the solution. However, the mass fraction of B₃O₃(OH)₄⁻ and B(OH)₃ can be neglected if (Na₂O) is more than 0.028. The dominant boron species is B(OH)₄⁻ in the solution if (Na₂O) is more than 0.028.

4. Conclusions

The Pitzer model of boron species in the ternary system Na₂O-B₂O₃-H₂O at 298.15 K was constructed. The solubilities; concentration of four boron species B(OH)₃, B(OH)₄⁻, B₃O₃(OH)₄⁻, and B₄O₅(OH)₄²⁻, and H⁺ (OH^–) in the system Na₂O-B₂O₃-H₂O were calculated with the Pitzer model. The boron species B₅O₆(OH)₄⁻ was not considered in the model calculation. The calculated solubility and pH data are in accordance with the experimental results. The Pitzer model can be used to describe the experimental values. The distributions of the four boron species in the solution were obtained. The mole fractions of the four boron species are mainly affected by the ratio of B₂O₃ and Na₂O in the solution. The dominant boron species in the solution saturated with different salts vary greatly. The main boron species are B(OH)₃ and B₃O₃(OH)₄^– in the solution saturated with B(OH)₃ or NaB₅O₆(OH)₄·3H₂O. B(OH)₄^– is the dominant boron species in the solution saturated with NaB(OH)₄ · 2H₂O. The results can supply a theoretical reference for sodium borate separation from brine.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was financially supported by the National Natural Science Foundation of China (22073068 and U1507112), Foundation of Tianjin Key Laboratory of Marine Resources and Chemistry (Tianjin University of Science & Technology) (2018-04), the Key Projects of Natural Science Foundation of Tianjin (18JCZDJC10040), and Yangtze Scholars and Innovative Research Team of the Chinese University (IRT-17R81).

References

M. L. Zhao, S. Q. Wang, X. N. Han, Y. F. Guo, and T. L. Deng, “Metastable equilibria of Li₂SO₄−LiBO₂−H₂O ternary system at 288.15 K,” Chemical Engineering, vol. 43, pp. 37–40, 2015, in Chinese.
View at: Publisher Site | Google Scholar
L. Yang, L. Chen, T. Zhang et al., “Solubility determination and thermodynamic modeling of solid−liquid equilibria in the LiBO₂−Li₂B₄O₇−H₂O system at 298.15 K and 323.15 K,” Fluid Phase Equilibria, vol. 523, Article ID 112783, 2020.
View at: Publisher Site | Google Scholar
S. Wang, J. Yang, C. Shi, D. Zhao, Y. Guo, and T. Deng, “Solubilities, densities, and refractive indices in the ternary systems (LiBO₂ + NaBO₂ + H₂O) and (LiBO₂ + KBO₂ + H₂O) at 298.15 K and 0.1 MPa,” Journal of Chemical & Engineering Data, vol. 64, no. 7, pp. 3122–3127, 2019.
View at: Publisher Site | Google Scholar
L. Li, Y. Guo, S. Zhang, M. Shen, and T. Deng, “Phase equilibria in the aqueous ternary systems (LiCl + LiBO₂ + H₂O) and (Li₂SO₄ + LiBO₂ + H₂O) at 323.15 K and 0.1 MPa,” Fluid Phase Equilibria, vol. 436, pp. 13–19, 2017.
View at: Publisher Site | Google Scholar
W. Y. Zhang, X. P. Li, L. Yang, and S. H. Sang, “Experimental study on phase equilibria in Na₂B₄O₇ + Na₂SO₄ + H₂O and Li₂B₄O₇ + Na₂B₄O₇ + H₂O aqueous ternary systems at 273 K,” Russian Journal of Physical Chemistry A, vol. 93, no. 6, pp. 1032–1037, 2019.
View at: Publisher Site | Google Scholar
P. C. Ma and P. H. Heng, “Li⁺, Na⁺//Cl⁻, B₄O₇^2– –H₂O reciprocal quaternary system solubility phase diagram,” Chemistry Journal Chinese University, vol. 397, pp. 77–83, 1988.
View at: Google Scholar
Y. Q. Zhou, C. H. Fang, Y. Fang, and F. Y. Zhu, “Density, electrical conductivity, acidity, viscosity and Raman spectra of aqueous NaBO₂, Na₂B₄O₇ and NaB₅O₈ solutions at 298.15 and 323.15 K,” Journal of the Chemical Society of Pakistan, vol. 35, no. 4, pp. 2771–2783, 1986.
View at: Google Scholar
H. W. Ge, C. H. Fang, Y. Fang et al., “Density, electrical conductivity, pH, and polyborate distribution of LiB(OH)₄, Li₂B₄O₅(OH)₄, and LiB₅O₆(OH)₄ solutions,” Journal of Chemical and Engineering Data, vol. 59, no. 11, pp. 4039–4048, 2014.
View at: Publisher Site | Google Scholar
C. H. Fang, Y. Fang, Y. Q. Zhou, F. Y. Zhu, L. H. Yan, and Z. W. Qian, “Recent progress on structure of aqueous polyborate solutions,” Journal of Salt Lake Research, vol. 27, no. 2, pp. 11–39, 2019.
View at: Publisher Site | Google Scholar
J. C. Zhang, M. Wang, and J. Dai, “Summarization of the lithium extraction system,” Journal of Salt Lake Research, vol. 13, no. 1, pp. 42–48, 2005, in Chinese.
View at: Google Scholar
L. Yang, L. Z. Meng, T. J. Ge, D. Li, T. L. Deng, and Y. F. Guo, “Solubility measurement and thermodynamic modeling of Solid−liquid equilibria in the MCl−M₂B₄O₇−H₂O (M = Li, Na) systems,” Journal of Chemical and Engineering Data, vol. 64, no. 9, pp. 4510–4517, 2019.
View at: Publisher Site | Google Scholar
М. Г. Валяшко and Г. К. Голе, “Solubility data in the system Na₂O–B₂O₃–H₂O at 25°C,” ЖНХ, vol. 5, no. 6, p. 1318, 1960.
View at: Google Scholar
N. S. Kurnakov, M. I. Ravich, and N. V. Troitskaya, “Solubility data in the system Na₂O–B₂O₃–H₂O at 25°C,” Sektora Fiz.-Khim, Anal., Akad.Nauk SSSR, vol. 10, p. 298, 1938.
View at: Google Scholar
W. Blasdale and C. Slansky, “The solubility curves of boric acid and the borates of sodium,” Journal of the American Chemical Society, vol. 61, p. 919, 1939.
View at: Publisher Site | Google Scholar
D. Howard and L. Silcock, Solubilities of Inorganic and Organic Compounds, Pergamon Press, New York, NY, USA, 1979.
T. Zhang, D. Li, and L. Z. Meng, “Recent progresses on the boron species in aqueous solution: structure, phase equilibria, metastable zone width(MZW) and thermodynamic model,” Reviews in Inorganic Chemistry, vol. 22, 2020, in press.
View at: Publisher Site | Google Scholar
P. Wang, J. J. Kosinski, M. M. Lencka, A. Anderko, and R. D. Springer, “Thermodynamic modeling of boric acid and selected metal borate systems,” Pure and Applied Chemistry, vol. 85, no. 11, pp. 2117–2144, 2013.
View at: Publisher Site | Google Scholar
D. Li, Y. Liu, L. Meng, Y. Guo, T. Deng, and L. Yang, “Phase diagrams and thermodynamic modeling of solid-liquid equilibria in the system NaCl-KCl-SrCl₂-H₂O and its application in industry,” The Journal of Chemical Thermodynamics, vol. 136, pp. 1–7, 2019.
View at: Publisher Site | Google Scholar
L. Yang, D. Li, T. Zhang, L. Meng, T. Deng, and Y. Guo, “Thermodynamic phase equilibria in the aqueous ternary system NaCl-NaBO₂-H₂O: experimental data and solubility calculation using the Pitzer model,” The Journal of Chemical Thermodynamics, vol. 142, Article ID 106021, 2020.
View at: Publisher Site | Google Scholar
A. R. Felmy and J. H. Weare, “The prediction of borate mineral equilibria in natural waters: application to Searles Lake, California,” Geochimica et Cosmochimica Acta, vol. 50, no. 12, pp. 2771–2783, 1986.
View at: Publisher Site | Google Scholar
L. L. Chen, X. X. Yang, Y. F. Guo, T. L. Deng, D. Li, and L. Z. Meng, “Solubility determination and thermodynamic modelling of solid-liquid equilibria in the (NaCl + NaBO₂ + Na₂B₄O₇ + H₂O) system at 298.15 K,” The Journal of Chemical Thermodynamics, vol. 152, Article ID 106283, 2020.
View at: Publisher Site | Google Scholar
D. Li, Q. T. Fu, T. Zhang et al., “Thermodynamic modeling of boron species in brine systems containing metaborate and its application in evaporation simulation,” Journal of Materials Research and Technology, vol. 9, no. 11, pp. 13067–13075, 2020.
View at: Publisher Site | Google Scholar
K. S. Pitzer, “Thermodynamics of electrolytes. I. Theoretical basis and general equations,” The Journal of Physical Chemistry, vol. 77, no. 2, pp. 268–277, 1973.
View at: Publisher Site | Google Scholar
K. S. Pitzer, “Ion interaction approach: theory and data correlation,” in Activity Coefficients in Electrolyte Solutions, CRC Press, Boca Raton, FL, USA, 2nd edition, 1991.
View at: Google Scholar
C. E. Harvie and J. H. Weare, “The prediction of mineral solubilities in natural waters: the Na—K—Mg—Ca—Cl—SO₄—H₂O system from zero to high concentration at 25°C,” Geochimica et Cosmochimica Acta, vol. 44, no. 7, pp. 981–997, 1980.
View at: Publisher Site | Google Scholar
C. E. Harvie, N. Møller, and J. H. Weare, “The prediction of mineral solubilities in natural waters: the Na-K-Mg-Ca-H-Cl-SO₄-OH-HCO₃-CO₃-CO₂-H₂O system to high ionic strengths at 25°C,” Geochimica et Cosmochimica Acta, vol. 48, no. 4, pp. 723–751, 1984.
View at: Publisher Site | Google Scholar
R. L. Li and H. Y. Ye, “Phase equilibrium of NaNO₃-H₃BO₃-H₂O ternary system at 25°C,” Inorganic Chemicals Indusrry, vol. 44, no. 6, pp. 31–33, 2012.
View at: Google Scholar

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