A universal thermodynamic model of calculating mass action concentrations forstructural units or ion couples in aqueous solutions and its applications in binary and ternary aqueous solutions

YANG Xue-min¹, ZHAO Wei-jie^{2, 3}, CHAI Guo-ming², GUO Han-jie², ZHANG Qiang³

1. State Key Laboratory of Multiphase Complex Systems, Institute of Process Engineering,

Chinese Academy of Sciences, Beijing 100190, China;

2. School of Metallurgical and Ecological Engineering,

University of Science and Technology Beijing, Beijing 100083, China;

3. Beijing Metallurgical Equipment Research & Design Corporation Ltd.,

China Metallurgical Group Corporation, Beijing 100029, China

Received 16 May 2010; accepted 28 November 2010

Abstract: A universal thermodynamic model of calculating mass action concentrations for structural units or ion couples in ternary and binary strong electrolyte aqueous solution was developed based on the ion and molecule coexistence theory and verified in four kinds of binary aqueous solutions and two kinds of ternary aqueous solutions. The calculated mass action concentrations of structural units or ion couples in four binary aqueous solutions and two ternary solutions at 298.15 K have good agreement with the reported activity data from literatures after shifting the standard state and concentration unit. Therefore, the calculated mass action concentrations of structural units or ion couples from the developed universal thermodynamic model for ternary and binary aqueous solutions can be applied to predict reaction ability of components in ternary and binary strong electrolyte aqueous solutions. It is also proved that the assumptions applied in the developed thermodynamic model are correct and reasonable, i.e., strong electrolyte aqueous solution is composed of cations and anions as simple ions, H₂O as simple molecule and other hydrous salt compounds as complex molecules. The calculated mass action concentrations of structural units or ion couples in ternary and binary strong electrolyte aqueous solutions strictly follow the mass action law.

Key words: universal thermodynamic model; mass action concentration; activity; ternary aqueous solution; binary aqueous solution; ion and molecule coexistence theory; structural units; ion couples; components

1 Introduction

The thermodynamic properties of strong electrolyte aqueous solutions, such as activity of component, have been widely applied to biological and industrial processes[1], hydrometallurgy, environmental chemistry[2-5], geochemistry and so on[6]. However, with respect to abundant kinds of electrolyte aqueous solutions, the activity data are not enough to fulfill practical application, or the correlations between activity coefficient and concentration for various solutions have several formulas. In addition, not only ions, but also molecules, such as hydrous salt molecules, can exist in strong electrolyte aqueous solutions. Therefore, it is not very easy to determine activities of components in strong electrolyte aqueous solutions by using traditional activity calculation methods. To accurately predicate activities of components with limited data, it is of importance to develop a universal method to calculate activity of components in strong electrolyte aqueous solutions.

According to the fact that not only ions but also molecules can be found in metallurgical slags or melts at high temperature, the ion and molecule coexistence theory (IMCT) proposed by ZHANG[7-9] has been successfully applied to calculate mass action concentrations of structural units or ion couples to describe reaction ability, like classical concept of activity, for various metallurgical slags and metal melts. The essential cores of IMCT[7-9] are summarized as follows: 1) ions, simple and complex molecules exist in a solution simultaneously as structural units; 2) there are dynamic equilibriums of chemical reactions between ion couples and simple molecules to form complex molecules in a solution; 3) simple ions take part in chemical reactions of forming complex molecules with simple molecules as ion couples, which are composed of cations and anions based on electrovalence balance principle; 4) the calculated equilibrium mole fractions of structural units or ion couples are defined as mass action concentration, which are similar to activities of corresponding components.

The mass action concentrations of structural units or ion couples in binary aqueous solutions of KCl-H₂O, CsCl-H₂O, NaCl-H₂O and BaCl₂-H₂O[10]; NaBr-H₂O, LiNO₃-H₂O, HNO₃-H₂O and KF-H₂O[11]; KBr-H₂O, NH₄Br-H₂O and ZnBr₂-H₂O[11]; and ternary aqueous solutions of NaCl-KCl-H₂O[13]; RbCl-RbNO₃- H₂O[14]; NaClO₄-NaF-H₂O[15] at 298.15 K have been successfully calculated by using the developed thermodynamic model of calculating mass action concentrations for structural units or ion couples from IMCT[7-9]. The results[10-15] show that the calculated mass action concentrations of structural units or ion couples in above-mentioned binary and ternary aqueous solutions have good agreement with the reported activities of corresponding components in literatures. Therefore, it is verified that the application scope of IMCT[7-9] can be expanded from metallurgical slag and metal melts at high temperature to electrolyte aqueous solutions at low or ambient temperature. However, a universal thermodynamic model for binary and ternary aqueous solutions has not been reported and summarized in previous publications[10-15].

In this work, a universal thermodynamic model of calculating mass action concentrations for structural units or ion couples in ternary electrolyte aqueous solutions with hydrous salt compounds as complex molecules was developed based on IMCT[7-9]. Meanwhile, a universal thermodynamic model for binary aqueous solutions with hydrous salt compounds as complex molecules was also developed by simplifying the universal thermodynamic model for ternary solutions, and four kinds of binary aqueous solutions were chosen to verify the developed thermodynamic model as: 1) binary aqueous solutions without any hydrous salt compound formed; 2) binary solutions with ECl?2H₂O as formed hydrous salt compound with unknown standard molar Gibbs free energy ; 3) binary aqueous solutions with ECl?2H₂O as formed hydrous salt compound with known ; 4) binary aqueous solutions with ECl?H₂O and ECl?3H₂O as formed hydrous salt compounds with known . All universal thermodynamic models were verified from the previous published data of calculating mass action concentrations for structural units or ion couples in binary and ternary solutions[10-15] as examples. The ultimate aims of this work are to develop a universal thermodynamic model of calculating mass action concentrations for structural units or ion couples in binary and ternary electrolyte aqueous solutions and verify it in various special cases.

2 Establishment of universal thermodynamic model

2.1 Hypotheses

According to the facts that there are ions and molecules simultaneously in strong electrolyte aqueous solutions, the assumptions applied in the developed universal thermodynamic model of calculating mass action concentrations for structural units or ion couples in ternary strong electrolyte aqueous solutions can be proposed based on IMCT[7-9] by taking ECl-FCl-H₂O as an example aqueous solution as follows.

1) The structural units in ECl-FCl-H₂O ternary solutions are composed of E⁺, F⁺ and Cl^- as simple ions, H₂O as simple molecule, ECl?H₂O, ECl?2H₂O, , ECl?nH₂O or FCl?H₂O, FCl?2H₂O, , FCl?mH₂O as complex molecules. Each cation or anion of the simple ions in ECl-FCl-H₂O ternary solutions occupies only one position of structural units, but will take part in reaction of forming hydrous salt molecules in the form of ion couple as (E⁺+Cl^-), and (F⁺+Cl^-). For example, ECl in ECl-FCl-H₂O ternary solutions can be electrolyzed or separated into two simple ions as E⁺ and Cl^- as two structural units, respectively; but ions of E⁺ and Cl^- will take part in reaction of forming hydrous salt molecules as an ion couple (E⁺+Cl^-) in ECl-FCl-H₂O ternary solutions.

2) Formation of complex hydrous salt molecules by ion couples and simple molecule of H₂O is under dynamic equilibrium as

(E⁺+Cl^-)+H₂O=ECl?H₂O                      (1.1)

(E⁺+Cl^-)+2H₂O=ECl?2H₂O                    (1.2)

(E⁺+Cl^-)+nH₂O=ECl?nH₂O                    (1.n)

(F⁺+Cl^-)+H₂O=FCl?H₂O                      (2.1)

(F⁺+Cl^-)+2H₂O=FCl?2H₂O                    (2.2)

(F⁺+Cl^-)+mH₂O=FCl?mH₂O                   (2.m)

3) The structural units in ECl-FCl-H₂O strong solutions bear continuity in the investigated concentration range of solutes.

4) All chemical reactions of forming complex hydrous salt molecules in ECl-FCl-H₂O strong electrolyte ternary aqueous solutions as shown in Eq.(1) and Eq.(2) follow the mass action law.

2.2 Universal thermodynamic model for ternary systems

2.2.1. Model for ternary systems with all hydrous salt molecules formed

Based on the above-mentioned hypotheses, the structural units in a chloride ternary aqueous solution with all hydrous salt molecules formed as an example, i.e., ECl-FCl-H₂O ternary solution, are composed of E⁺, F⁺ and Cl^- as simple ions, H₂O as simple molecule, ECl?H₂O, ECl?2H₂O, , ECl?nH₂O, and FCl?H₂O, FCl?2H₂O, , FCl?mH₂O as complex molecules or hydrous salt molecules.

The morality  (mol/kg (H₂O)) is usually applied to represent concentration of components in strong electrolyte aqueous solution, i.e., mole number of solutes in 1 kg solvent of H₂O. Therefore, mole number of ECl, FCl and H₂O in ECl-FCl-H₂O ternary solution based on 1 kg H₂O before equilibrium is defined as  and  respectively. The mole fractions of ECl, FCl and H₂O in ECl-FCl-H₂O ternary solution before equilibrium can be easily

determined as

The mole numbers of structural units in ECl-FCl-H₂O ternary solution under dynamic reaction equilibrium condition based on 1 kg H₂O are defined according to IMCT[7-9] as follows:

,

,

                                 (3)

Hence, the total mole number of all structural units  in ECl-FCl-H₂O ternary solution under equilibrium condition based on 1 kg H₂O can be calculated according to the mass balance principle as

       (4)

The mass action concentration of structural unit is defined as a ratio of equilibrium mole number of structural unit i to the total equilibrium mole number of all structural units in a system with a fixed amount according to IMCT[7-9], and can be calculated by

                               (-) (5)

It should be emphasized that mass action concentrations of all structural units in forms of ions, simple and complex molecules can be calculated from Eq.(5) directly[7-9]; however, mass action concentration of an ion couple from cation and anion, such as (E⁺+Cl^-), should be calculated as follows[7-9]:

       (-) (6)

Therefore, the mass action concentrations of structural units or ion couples in ECl-FCl-H₂O ternary solution under equilibrium condition can be expressed as follows:

(-) (7)

The possible dynamic reactions of forming various hydrous salt molecules and their corresponding equilibrium constants and mass action concentrations can be describled as follows:

(E⁺+Cl^-)+H₂O=ECl?H₂O,

                  (8.1)

(E⁺+Cl^-)+2H₂O=ECl?2H₂O,

,               (8.2)

(E⁺+Cl^-)+nH₂O=ECl?nH₂O,

                 (8.n)

(F⁺+Cl^-)+H₂O=FCl?H₂O,

                  (9.1)

(F⁺+Cl^-)+2H₂O=FCl?2H₂O,

,                  (9.2)

(F⁺+Cl^-)+mH₂O=FCl?mH₂O,

               (9.m)

The mass balance of three components as ECl, FCl and H₂O in ECl-FCl-H₂O ternary solution based on 1 kg H₂O according to IMCT[7-9] can be represented as follows:

(10a)

      (10b)

 (10c)

Combining Eq.(10a)-Eq.(10b) to eliminate , the following formulations can be obtained as

        (11a)

    (11b)

According to the fact that the total equilibrium mole fraction of all structural units in a system with a fixed amount is 1.0, the following equation can be obtained as

   (-) (12)

The equilibrium constant  of all dynamic chemical reactions describled in Eq.(1) and Eq.(2) can be calculated as follows:

                    (-) (13)

Therefore, the equation groups of Eq.(10) and Eq.(12) or Eq.(11) and Eq.(12) are composed of the universal thermodynamic model of calculation mass action concentrations for structural units or ion couples  in ECl-FCl-H₂O ternary aqueous solution, respectively. There are four unknown parameters as , ,  and  with four independent equations in the universal thermodynamic model composing of Eq.(10) and Eq.(12), or three unknown parameters as N₁, N₂ and N₃ with three independent equations in the universal thermodynamic model composing of Eq.(11) and Eq.(12). It is clear that the real solutions of above-mentioned mass action concentrations of structural units or ion couples N_i and total mole fraction of all structural units  in ECl-FCl-H₂O ternary aqueous solution can be solely solved by combining Eq.(10) and Eq.(12) or Eq.(11) and Eq.(12) under condition of knowing equilibrium constant . The equilibrium mole number of each structural unit n_i can be also determined by Eq.(5) or Eq.(6) after knowing N_i and .

The calculated mass action concentration  is based on pure mater as standard state and mole fraction x_i as concentration unit; however, the reported activity data of strong electrolyte aqueous solutions in literatures usually choose infinite dilute solution as standard state and molality m_i (mol/kg (H₂O)) as concentration unit. To compare the calculated mass action concentrations N_i and reported activity data a_i, a transformation coefficient L_i for solvent or  for solutes should be introduced to shift standard state and concentration unit[10-15], and the transformed mass action concentration of components  can be calculated as follows[10-15]:

      (-) (14)

It should be specially pointed out that mass action concentration of all ion couples, simple and complex (hydrous salts) molecules, such as ECl, FCl, H₂O, ECl?H₂O, ECl?2H₂O, , ECl?nH₂O, and FCl?H₂O, FCl?2H₂O, , FCl?mH₂O in ECl-FCl-H₂O ternary solution under equilibrium condition can be calculated from the developed universal thermodynamic model based on IMCT[7-9]. However, only activity data of three components in ECl-FCl-H₂O ternary solution as ECl, FCl and H₂O can be determined from viewpoints of classical experimental tests and traditional thermodynamics. Although concept of mass action concentration of components N_i, taking ECl as an example, has been applied in previous publications[7-15] and this study, the real meaning of mass action concentration for ECl N_ECl is the sum of mass action concentration of two contained structural units in ECl as E⁺ and Cl^-. From absolute viewpoint of IMCT[10-15], there is no mass action concentration of ECl as an independent component, because no independent ECl molecules can exist in ECl-FCl-H₂O ternary solution. Adopting concept of mass action concentration for ECl is just for convenient comparison of the reported ECl activity data with calculated mass action concentration of the same components.

2.2.2 Model for ternary systems without any hydrous salt molecule formed

As an extreme example, a universal thermodynamic model of calculating mass action concentrations for structural units or ion couples in ECl-FCl-H₂O ternary solution without any complex hydrous salt molecule formed can be also obtained by considering n, m and  as zero simultaneously in the equation groups of Eq.(10) and Eq.(12) or Eq.(11) and Eq.(12) choosing ECl-FCl-H₂O ternary solution as a representative.

The mole numbers of ECl, FCl and H₂O in ECl-FCl-H₂O ternary solution before reaction equilibrium based on 1 kg H₂O are represented as   =55.6 mol, respectively. The structural units in ECl-FCl-H₂O ternary solution without any hydrous salt molecule formed are composed of E⁺, F⁺ and Cl^- as simple ions, H₂O as simple molecules. The mole numbers of structural units in ECl-FCl-H₂O ternary solution without any hydrous salt molecule formed under dynamic reaction equilibrium condition based on 1 kg H₂O can be expressed as    Hence, the total mole number of all structural unitsbased on 1 kg H₂O in ECl-FCl-H₂O ternary solution without any hydrous salt molecule formed under equilibrium condition can be calculated according to the mass balance principle as

        (15)

The mass balance of ECl, FCl and H₂O in ECl-FCl-H₂O ternary solution can be described by considering definition of mass action concentrations of structural units or ion couples in ECl-FCl-H₂O ternary solution shown in Eq.(10) as

             (16a)

            (16b)

        (16c)

The following equations can be obtained by combining Eq.(16a)-Eq.(16c) to eliminate  as

                           (17a)

                          (17b)

The following expression can be obtained according to the principle that the sum of all mole fraction of structural units in ECl-FCl-H₂O ternary solution with a fixed amount under equilibrium condition is equal to 1.0, as

                            (18)

Therefore, the thermodynamic model of calculating mass action concentrations for structural units or ion couples in ECl-FCl-H₂O ternary solution without any hydrous salt molecule formed can be described by equation groups of Eq.(16) and Eq.(18) or Eq.(17) and Eq.(18). Substituting the known parameters as  and b₃= =55.6 mol into Eq.(17)-Eq.(18), the thermodynamic model of calculating mass action concentrations for structural units or ion couples in ECl-FCl-H₂O ternary solution without any hydrous salt molecule formed can be simplified as

              (19)

The transformation coefficient L_i or and transformed mass action concentration of components  can be obtained by Eq.(14) as described in Section 2.2.1.

2.3 Universal thermodynamic model for binary systems

2.3.1 Model for binary systems with all hydrous salt molecules formed

The structural units in ECl-H₂O binary strong electrolyte aqueous solution with all hydrous salt molecules formed are composed of E⁺ and Cl^- as simple ions, H₂O as simple molecule and ECl?H₂O, ECl?2H₂O, , and ECl?nH₂O as complex hydrous salt molecules. Therefore, the mole numbers of ECl and H₂O in ECl-H₂O binary solution based on 1 kg H₂O before equilibrium is defined asand b₃= respectively. The mole fractions of ECl and H₂O in ECl-H₂O binary solution before equilibrium can be easily determined by

The mole number of structural units in ECl-H₂O binary solution under dynamic reaction equilibrium condition based on 1 kg H₂O is defined according to IMCT[7-9] as follows:

,              (20)

Hence, the total mole number of all structural units  based on 1 kg H₂O in ECl-H₂O binary solution under equilibrium condition can be calculated according to the mass balance principle as

           (21)

Therefore, Eq.(10) can be rewritten to present the mass balance of ECl and H₂O in ECl-H₂O binary solution with all hydrous salt molecules formed under equilibrium condition based on 1 kg H₂O by ignoring parameters b₂, m and  as

     (22a)

      (22b)

The following equation can be obtained by combining Eq.(22a) and Eq.(22b) to eliminate  as

   (23)

According to the fact that the total equilibrium mole fraction of all structural units in a system with a fixed amount is 1.0, Eq.(12) can be simplified for ECl-H₂O binary solution as

(-) (24)

Therefore, Eq.(22) and Eq.(24) or Eq.(23) and Eq.(24) are the developed universal thermodynamic model of calculating mass action concentrations for structural units or ion couples N_i in ECl-H₂O binary solution with all hydrous salt molecules formed, respectively. There are three unknown parameters as N₁, N₃ and  with three independent equations in the universal thermodynamic model composing of Eq.(22) and Eq.(24), or two unknown parameters as N₁ and N₃ with two independent equations in the universal thermodynamic model composing of Eq.(23) and Eq.(24). The real solutions of N_i and  for ECl-H₂O binary solution can be solely solved by combing Eq.(22) and Eq.(24) or Eq.(23) and Eq.(24) under condition of knowing . The equilibrium mole number of all structural units or ion couples  under equilibrium condition can be also calculated from definition formula of N_i in Eq.(5) or Eq.(6) after knowing N_i and . The expressions of  L_i or  and  for ECl-H₂O binary solution are the same as equations described in Eq.(13) and Eq.(14).

2.3.2 Model for binary system without any hydrous salt molecule formed

The structural units in ECl-H₂O binary strong electrolyte aqueous solution without any hydrous salt complex molecule generated are composed of E⁺ and Cl^- as simple ions, and H₂O as simple molecule. Hence, no equilibrium constant of forming hydrous salt complex molecules  can exist in the universal thermodynamic model of calculating mass action concentrations for structural units or ion couples N_i in ECl-H₂O binary solution, as shown in Eq.(22) and Eq.(24) or Eq.(23) and Eq.(24), respectively. The mole numbers of ECl and H₂O in ECl-H₂O binary solution without any hydrous salt complex molecule generated before reaction equilibrium based on 1 kg H₂O are presented as   55.6 mol. The mole numbers of structural units in ECl-H₂O ternary solution without any hydrous salt molecule formed under dynamic reaction equilibrium condition based on 1 kg H₂O can be expressed[7-9]   as  55.6. Hence, the total mole number of all structural units  based on 1 kg H₂O in ECl-H₂O ternary solution without any hydrous salt molecule formed under equilibrium condition can be calculated according to the mass balance principle as

                  (25)

The mass balance of ECl and H₂O in ECl-H₂O binary solution without any hydrous salt molecule formed under equilibrium condition based on 1 kg H₂O can be presented by simplifying Eq.(22) as

             (26a)

       (26b)

The following equation can be obtained by combining Eq.(26a) and Eq.(26b) to eliminate , as

                           (27)

According to the fact that the total equilibrium mole fraction of all structural units in a system with a fixed amount is 1.0, Eq.(24) can be simplified for ECl-H₂O binary solution without any complex hydrous salt molecule formed as

                             (-) (28)

The equation groups of Eq.(26) and Eq.(28) or Eq.(27) and Eq.(28) are the thermodynamic model of calculating mass action concentrations for structural units or ion couples N_i in ECl-H₂O binary solution without any complex hydrous salt molecule formed, respectively. N_i can be calculated by solving Eq.(26) and Eq.(28) or Eq.(27) and Eq.(28) as follows:

                    (29)

The expressions of L_i or  and  for ECl-H₂O binary solution without any complex hydrous salt molecule formed are the same as equations described in Eq.(14).

2.3.3 Model for binary system with ECl?2H₂O as formed hydrous salt molecule with unknown

In the case of no standard molar Gibbs free energy  of the existed complex hydrous salt molecules in ECl-H₂O binary solution can be found from related literatures, the common procedure of developing the thermodynamic model of calculating mass action concentrations for structural units or ion couples  from IMCT[7-9] is summarized as follows: 1) confirming the chemical formulas of possibly formed hydrous salt molecules from phase diagram of the studied ECl-H₂O binary solution; 2) establishing the thermodynamic model from the proposed universal thermodynamic model; 3) substituting the mass action concentrations N_i of ECl and H₂O in ECl-H₂O binary solution by the reported activities of ECl and H₂O in formulas of equilibrium constant  for reactions of possibly formed hydrous salt molecules. When a relatively stable equilibrium constant  is obtained for the reaction of formed hydrous salt molecule, the thermodynamic model can be solved by using the calculated  to determine mass action concentrations of structural units or ion couples. Taking ECl?2H₂O as formed hydrous salt molecule in ECl-H₂O binary solution with unknown  as an example, the procedure of developing thermodynamic model is explained in the following text.

Certainly, the structural units in ECl-H₂O binary solution with ECl?2H₂O as hydrous salt molecule are E⁺ and Cl^- as simple ions, H₂O as simple molecule and ECl?2H₂O as complex hydrous salt molecule. The mole numbers of ECl and H₂O in ECl-H₂O binary solution with ECl?2H₂O as formed hydrous salt molecule before reaction equilibrium based on 1 kg H₂O are presented as , =55.6 mol. The mole numbers of structural units in ECl-H₂O ternary solution with ECl?2H₂O as formed hydrous salt molecule formed under dynamic reaction equilibrium condition based on 1 kg H₂O can be expressed as 55.6, Hence, the total mole number of all structural units  in ECl-H₂O binary solution with ECl?2H₂O as formed hydrous salt molecules under equilibrium condition based on 1 kg H₂O can be calculated according to the mass balance principle as

                        (30)

The mass balance of ECl and H₂O in ECl-H₂O binary solution based on 1 kg H₂O by Eq.(22) can be rewritten as

   (31a)

           (31b)

The developed universal thermodynamic model of ECl-H₂O binary solution shown in Eq.(23) and Eq.(24) can be simplified as

          (32)

                       (33)

The chemical reaction equilibrium constant  of reaction for forming ECl?2H₂O by Eq.(1.2) can be presented by considering Eq.(32) or Eq.(33), respectively, as

                   

                   (34a)

                       (34b)

It should be specially pointed out that N_i is calculated based on pure matter as standard state and mole fraction x_i as concentration unit, which is the same as the reported  from literatures. Usually, activity of H₂O, , but not activity coefficient of H₂O, is reported in related literatures. Unfortunately, only activity coefficient not a_i, is reported from literatures for solute i, which is based on infinite dilute solution as standard state and molality m_i as concentration unit. Hence, only  can be applied to replace , while, a_ECl, i.e., product of the reported activity coefficient  and m_ECl, cannot be used to substitute for N_ECl because N_ECl is less than 1.0 while a_ECl can be much larger than 1.0 when  is high enough. Under the circumstances, the calculated  from multiplying the reported  by x_ECl, which is less than 1.0, is recommended to substitute N_ECl in Eq.(34a) or Eq.(34b)[7-9]. However, the calculated  from Eq.(34a) is certainly not equal to that from Eq.(34b) because Eq.(34a) is formulated from mass balance of solute i and solvent H₂O, while Eq.(34b) is based on mole fraction of all structural units in a close system as 1.0 by using N_i as concentration unit. Therefore, the reported  from literature and product of the reported  and x_ECl as  are applied in Eq.(34a) to determine  according to IMCT[7-9].

When the calculated  from Eq. (34a) by using reported  and  from literatures is constant with a very narrow fluctuation in various x_ECl or m_ECl, the calculated  can be applied in the developed thermodynamic model as shown in equation groups of Eq.(31) and Eq.(33) or Eq.(32) and Eq.(33) to solve N_i,  and n_i. The expressions of L_i or  and for ECl-H₂O binary solution with ECl?2H₂O as complex hydrous salt molecule are the same as equations described in Eq.(14).

2.3.4 Model for binary system with ECl?2H₂O as formed hydrous salt molecule with known

The structural units in ECl-H₂O binary solution with ECl?2H₂O as formed hydrous salt molecule with known  consist of E⁺ and Cl^- as simple ions, H₂O as simple molecule and ECl?2H₂O as complex molecule. The mole numbers of ECl and H₂O in ECl-H₂O binary solution with ECl?2H₂O as formed hydrous salt molecule before reaction equilibrium based on 1 kg H₂O are presented as , =55.6 mol. The mole number of structural units in ECl-H₂O binary solution with ECl?2H₂O as formed hydrous salt molecule under dynamic reaction equilibrium condition based on 1 kg H₂O can be expressed as  =55.6, n₄₂= Hence, the total mole number of all structural units in ECl-H₂O binary solution with ECl?2H₂O as formed hydrous salt molecule under equilibrium condition based on 1 kg H₂O can be calculated according to the mass balance principle as

                        (35)

The mass balance of ECl and H₂O in ECl-H₂O binary solution based on 1 kg H₂O can be presented by Eq.(31) or Eq.(32), and equilibrium constant  can be calculated from by Eq.(13).

Therefore, the thermodynamic model of ECl-H₂O binary solution with ECl?2H₂O as formed hydrous salt with known  is presented by combining of Eq.(31) and Eq.(33) or Eq.(32) and Eq.(33). Therefore, parameters as N_i,  and n_i in ECl-H₂O binary solution with ECl?2H₂O as formed hydrous salt with known  can be solved. The expressions of L_i or  and  for ECl-H₂O binary solution with ECl?2H₂O as complex hydrous salt molecule with known  are the same as equations described in Eq.(14).

2.3.5 Model for binary system with ECl?H₂O and ECl?3H₂O as formed hydrous salt molecule with known

In the case of forming ECl?H₂O and ECl?3H₂O as hydrous salt molecules with known , the structural units in ECl-H₂O binary solution are composed of E⁺ and Cl^- as simple ions, H₂O as simple molecule, ECl?H₂O and ECl?3H₂O as complex molecules. The mole numbers of ECl and H₂O in ECl-H₂O binary with ECl?H₂O and ECl?3H₂O as formed hydrous salt molecules before reaction equilibrium based on 1 kg H₂O is presented as  =55.6 mol. The mole number of structural units in ECl-H₂O binary solution with ECl?H₂O and ECl?3H₂O as formed hydrous salt molecules under dynamic reaction equilibrium condition based on 1 kg H₂O can be expressed as  =55.6, Hence, the total mole number of all structural units  based on 1 kg H₂O in ECl-H₂O binary solution with ECl?H₂O and ECl?3H₂O as formed hydrous salt molecules under equilibrium condition can be calculated according to the mass balance principle as

                    (36)

The mass balance of ECl and H₂O in ECl-H₂O binary solution with ECl?H₂O and ECl?3H₂O as formed hydrous salt molecules with known  can be presented by simplifying Eq.(22) as

        (37a)

  (37b)

Eq.(23) and Eq.(24) can be rewritten for ECl-H₂O binary solution with ECl?H₂O and ECl?3H₂O as formed hydrous salt molecules with known  as follows:

                       (38)

              (39)

The thermodynamic model of ECl-H₂O binary solution with ECl?H₂O and ECl?3H₂O as formed hydrous salt molecules with known  is represented by combining Eq.(37) and Eq.(39) or Eq.(38) and Eq.(39). The parameters as N_i,  and n_i can be solved by combining Eq.(37) and Eq.(39) or Eq.(38) and Eq.(39). The equilibrium constants of  and , L_i or  and  for ECl-H₂O binary solution with ECl?H₂O and ECl?3H₂O as formed hydrous salt molecules with known  are also the same as equations described in Eq.(13) and Eq.(14), respectively.

3 Application of universal thermodynamic model

3.1 Application of developed models in binary aqueous solutions

3.1.1 Model application in binary aqueous solutions without any hydrous salt molecule formed

The representatives of ECl-H₂O binary solutions without any hydrous salt complex molecule generated are KCl-H₂O, CsCl-H₂O, KBr-H₂O and NH₄Br-H₂O according to their phase diagrams[16]. Hence, the mass action concentrations of structural units or ion couples in KCl-H₂O[10], CsCl-H₂O[10], KBr-H₂O[12] and NH₄Br-H₂O[12] binary aqueous solutions at 298.15 K have been calculated by the developed thermodynamic model of calculating mass action concentrations for structural units or ion couples in ECl-H₂O binary solution without any hydrous salt complex molecule generated. Taking KCl-H₂O[10] binary solution as an example, the application of the developed universal thermodynamic model is briefly demonstrated hereinbelow.

The mass action concentrations of KCl and H₂O in KCl-H₂O binary solution without any hydrous salt molecule formed at 298.15 K can be calculated by Eq.(29) directly. The transformation coefficients  and  in various molalities of KCl are listed in Table 1, the relationships between  and reported activity[17] a_KCl against x_KCl,  and reported [17] against , are illustrated in Fig.1, respectively.

Fig.1 Comparison of calculated mass action concentrations  and with measured activities of KCl and H₂O as a_KCl  and  in Ref.[17] for KCl-H₂O binary solution without forming any complex hydrous salt molecule at 298.15 K

It can be seen from Table 1 that the transformation coefficients of KCl () keep constant with a small fluctuation range with an average datum of 18.099; meanwhile,  is also stable with 1.000 as average datum. It can be also observed from Fig.1 that  and  are in good agreement with the report in Ref.[17] a_KCl  and . Therefore, it can be suggested that the calculated mass action concentrations can be applied to present reaction ability of KCl and H₂O, and have a close corresponding relation with reported activity data; all hypotheses used

Table 1 Transformation coefficients between calculated mass action concentrations and reported activities of structural units or ion couples for investigated six strong electrolyte aqueous solutions with various molalities of solutes at 298.15 K and equilibrium constant of forming NaCl?2H₂O in NaCl-H₂O binary solution at 298.15 K

in the developed thermodynamic model are reasonable and can depict the intrinsic structure of KCl-H₂O binary aqueous solution.

3.1.2 Model application in binary system with ECl?2H₂O as formed hydrous salt molecule with unknown

NaCl-H₂O is a representative of ECl-H₂O binary aqueous solution with ECl?2H₂O as hydrous salt molecule with unknown  based on its phase diagram[18]. The developed thermodynamic model of calculating mass action concentrations for structural units or ion couples in ECl-H₂O binary solution with ECl?2H₂O as formed hydrous salt molecule without  has been successfully applied in NaCl-H₂O binary solution[10] at 298.15 K by using the determined stable equilibrium constant  through substituting  in formula of  by product of reported [17] and x_NaCl as a_NaCl and replacing  in formula of  by the reported respectively. The determined  has an average datum as 1.043 with a small fluctuation in various molalities of NaCl in NaCl-H₂O binary solution as shown in Table 1 at 298.15 K. The transformation coefficient  and  are stable with 58.897 and 0.960 as average datum, respectively, as shown in Table 1.

The transformed mass action concentrations  and  have been compared with the reported activities[17], a_NaCl  and  in Fig.2 in various mole fractions of NaCl and H₂O before equilibrium, respectively. There is a good corresponding relation between  and a_NaCl,  and  with small deviation, respectively. Therefore, NaCl?2H₂O surely exists in NaCl-H₂O binary solution as described in phase diagram of NaCl-H₂O binary solution[18], the developed thermodynamic model of NaCl-H₂O binary solution can represent the intrinsic structure of NaCl-H₂O binary solution.

3.1.3 Model application in binary system with ECl?2H₂O as formed hydrous salt molecule with known

NaBr-H₂O, KF-H₂O and ZnBr₂-H₂O are representatives of ECl-H₂O binary aqueous solution with ECl?2H₂O as hydrous salt molecule with known  from their phase diagrams[16]. The established thermodynamic model of calculating mass action concentrations for structural units or ion couples in ECl-H₂O binary solution with ECl?2H₂O as formed hydrous salt molecule with known  has been successfully applied in NaBr-H₂O and KF-H₂O[11], and ZnBr₂-H₂O[12] binary aqueous solutions at 298.15 K. Taking KF-H₂O[11] binary solution as an example to demonstrate the thermodynamic model of binary solution with KF?2H₂O as formed hydrous salt molecule with known  at 298.15 K, the calculated  in various molalities of KF has an average datum of 1611.0 as summarized in Table 1.

The transformed mass action concentration  and reported activity[19] a_KFare compared in Fig.3. There is a good corresponding relation between  and a_KF  with a small deviation. It can be seen from Table 1 that the transformation coefficient of KF () are relatively constant with 1611.0 as an average value. Hence, the hypotheses in the developed thermodynamic model of KF-H₂O binary system are reasonable; the calculated mass action concentration of KF can represent reaction ability of KF in KF-H₂O binary system; KCl?2H₂O surely exists in KF-H₂O binary solution as described in phase diagram of KF-H₂O binary solution[16]; the established thermodynamic model of KCl-H₂O binary solution can

Fig.2 Comparison of calculated mass action concentrations  and with measured activities of NaCl and H₂O as a_NaCl  and  in Ref.[17] for NaCl-H₂O binary solution with NaF?2H₂O as formed hydrous salt molecule at 298.15 K

Fig.3 Comparison of calculated mass action concentration  with measured activity of KF as a_KF  in Ref.[19] for KF-H₂O binary solution with KF?2H₂O as formed hydrous salt molecule at 298.15 K

represent the intrinsic structural properties of KCl-H₂O binary solution.

3.1.4 Model application in binary system with ECl?H₂O and ECl?3H₂O as formed hydrous salt molecules with known

HNO₃-H₂O is a representative of ECl-H₂O binary aqueous solution with ECl?H₂O and ECl?3H₂O as formed hydrous salt molecules with known from its phase diagram[16]. The developed thermodynamic model of calculating mass action concentrations for structural units or ion couples in ECl-H₂O binary with ECl?H₂O and ECl?3H₂O as formed hydrous salt molecules with known  has been successfully applied in HNO₃-H₂O binary solution at 298.15 K as reported in previous publication[11]. The calculated  has a narrow deviation with an average of 80 753.0 in large molality range as listed in Table 1. The transformed mass action concentrations of HNO₃  and reported activity[19]   with mole fraction before equilibrium  as horizontal axis are compared in Fig.4. There is a good corresponding relationship between  and  with a small deviation. It can be deduced from Fig.4 that the structural properties of considering HNO₃?H₂O and HNO₃?3H₂O as formed hydrous salt molecules in HNO₃-H₂O binary solution from phase diagram[16] are reasonable.

3.2 Application of developed models in ternary aqueous solutions

3.2.1 Model application in ternary systems with formed hydrous salt molecules

The developed thermodynamic model of calculating mass action concentrations for structural units or ion

Fig.4 Comparison of calculated mass action concentration with measured activity of HNO₃   in Ref.[19] for HNO₃-H₂O binary solution with HNO₃?H₂O and HNO₃?3H₂O as formed hydrous salt molecules at 298.15 K

couples in ternary aqueous solution has been successfully applied in NaCl-KCl-H₂O[13] and NaClO₄-NaF-H₂O [15] ternary solutions with formed hydrous salt molecules at 298.15 K. Choosing NaCl-KCl-H₂O ternary solution with NaCl·2H₂O as hydrous salt molecule at 298.15 K as an example, the structural units in NaCl-KCl-H₂O ternary solution are composed of Na⁺, K⁺, and Cl^- as simple ions, H₂O as simple and NaCl·2H₂O as complex molecules from the phase diagram[18]. The mass balance of NaCl, KCl and H₂O in NaCl-KCl-H₂O ternary solution based on 1 kg H₂O shown in Eq.(10) can be simplified as

   (40a)

            (40b)

              (40c)

Eq.(11) and Eq.(12) can be rewritten for NaCl-KCl-H₂O ternary solution as follows:

        (41a)

             (41b)

                 (42)

Therefore, the equation groups of Eq.(40) and Eq.(42) or Eq.(41) and Eq.(42) are composed of the universal thermodynamic model of calculation mass action concentrations for structural units or ion couples N_i in NaCl-KCl-H₂O ternary solution, respectively. The mass action concentrations of NaCl, KCl and H₂O in NaCl-KCl-H₂O ternary solution have been calculated and reported in detail in previous publication[13] in different mole number ratios of NaCl to KCl, i.e., r=m_NaCl/m_KCl, as 0.2, 0.5, 1.0, 2.0, 3.0, and 4.0, respectively, under condition of knowing  as describled in Section 3.1.2 as 1.042[13]. The transformed mass action concentration andhave been compared with the reported activities[20], a_NaCl a_KCl  and , with different r.

As a representative of results, the transformation coefficients  and  with r of 4.0 are listed in Table 1 as 55.606 and 16.157, respectively; meanwhile, comparison of  and  with the reported activities[20], a_NaCl  a_KCl  and , is also given in Fig.5 with r of 4.0, respectively.

It can be observed from Fig.5 that the transformed mass action concentrations  and , are in good agreement with the reported activities[20], a_NaCl  a_KCl  and  with r of 4.0. It is shown from Table 1 that  and  remain constant with small deviations with r of 4.0. Hence, the developed thermodynamic model of calculating mass action concentrations for structural units or ion couples in NaCl-KCl-H₂O ternary solution can be successfully applied to predict reaction ability and reflect structural characteristics.

3.2.2 Model application in ternary systems without any hydrous salt molecule formed

The developed thermodynamic model for ternary solution without any hydrous salt molecule formed has been successfully applied in RbCl-RbNO₃-H₂O ternary solution[14] at 298.15 K. Certainly, the structural units in RbCl-RbNO₃-H₂O ternary solution at 298.15 K are composed of Rb⁺, Cl^- and  as simple ions, H₂O as simple molecule from the phase diagram[18]. The mass balance of RbCl, RbNO₃ and H₂O in RbCl-RbNO₃-H₂O ternary solution based on 1 kg H₂O shown in Eq.(10) can be simplified as

            (43a)

        (43b)

Fig.5 Comparison of calculated mass action concentrations  and  with reported activities of NaCl, KCl and H₂O as a_NaCl  a_KCl  and in Ref.[20] for NaCl-KCl-H₂O ternary system with r of 4.0 at 298.15 K

       (43c)

Eq.(11) and Eq.(12) can be expressed for RbCl-RbNO₃-H₂O ternary solution, respectively, as

                          (44a)

                         (44b)

                     (45)

Therefore, the equation groups of Eq.(43) and Eq.(45) or Eq.(44) and Eq.(45) are composed of the universal thermodynamic model of calculation mass action concentrations for structural units or ion couples N_i in RbCl-RbNO₃-H₂O ternary solution, respectively. The mass action concentrations of RbCl, RbNO₃ and H₂O in RbCl-RbNO₃-H₂O ternary solution have been calculated and reported in detail in previous publication[14] in a range of the total ionic strength, i.e., , from 0.1 to 0.9 mol/kg (H₂O) in 0.1 mol/kg (H₂O) step with different ionic strength fractions of NaClO₄,  as 0, 0.2, 0.4, 0.5, 0.6 and 0.8, respectively[14]. The transformed mass action concentrations  and  have been compared with the reported activities[21], a_RbCl and  with different I values reported elsewhere[14].

As a representative of results, the transformation coefficients  and  with I of 0.5 mol/kg (H₂O) are listed in Table 1, meanwhile, comparison of  and  with the reported activities[21],  a_RbCl  and  , are also given in Fig.6 under condition of I of 0.5 mol/kg (H₂O), respectively.

It can be obtained from Table 1, Fig.6 and other results described in previous publication elsewhere[14] that  and  are in good agreement of reported activities[21], a_RbCl  and   in a large change range of I ; the transformation coefficients,  and , listed in Table 1, keep constant with 17.29 and 15.61 as average datum with small deviations under condition of I as 0.50 mol/kg (H₂O) for various , respectively. Therefore, the developed thermodynamic model of calculation mass action concentrations for structural units or ion couples in RbCl-RbNO₃-H₂O ternary solution can be successfully applied to predict reaction ability and reflect structural characteristics.

4 Discussion and error analysis

Although the transformed mass action concentrations of structural units or ion couples  in various binary and ternary aqueous solutions have good consistency with the reported activity a_i , small difference between  and a_i can be observed for some solutions in high concentration range of solute. The

Fig.6 Comparison of mass action concentrations  and  with reported activities of RbCl and NbNO₃ as a_RbCl and  in Ref.[21] for RbCl-RbNO₃-H₂O ternary solution with total ionic strength I of 0.5 mol/kg (H₂O) at 298.15 K

reasons of small errors of the transformed mass action concentrations of structural units or ion couples with the reported activities can be summarized as follows:

1) Solute in high concentration range almost reaches saturation.

2) A transformation coefficient  must be used when comparing the calculated mass action concentration N_i and the reported activity a_i  of solute i. However, the calculated  is based on pure matter as standard state and mole fraction  as concentration unit, while the reported a_i  is relative to infinite dilution as standard state and molality m_i as concentration unit. The average transformation coefficient  is the average of ratios for a_i to N_i with various solute concentrations in the investigated concentration range, some errors can be certainly generated from the average calculated transformation coefficient . The idea method is to compare the calculated mass action concentrations of structural units or ion couples and reported activities relative to pure matter as standard state and mole fraction x_i as concentration unit for solutes.

3) To some strong electrolyte aqueous solutions, equilibrium constant  of reactions for forming hydrous salt molecules cannot be accurately calculated because its free formation energy  cannot be found from related literatures. The suggested method to get  by ZHANG[7-9] is calculating  by substituting  in formula of  by product of reported activity coefficient f′_i multiplying by x_i as a′_i  for solute, but not traditional activity a_i  for solute, with a relatively stable in an acceptable fluctuation range. It is not an absolutely correct, but an acceptable method. The average datum  can certainly induce some deviations.

5 Conclusions

1) A universal thermodynamic model of calculating mass action concentrations for structural units or ion couples in ternary and binary strong electrolyte aqueous solutions based on the ion and molecular coexistence theory has been developed and verified in four kinds of binary and two kinds of ternary aqueous solutions. The calculated mass action concentrations of structural units or ion couples in four binary and two ternary solutions have good agreement with the reported activity data from literatures after shifting the standard state and concentration unit.

2) The mass action concentrations of structural units or ion couples calculated from the developed universal thermodynamic model for calculating mass action concentrations of structural units or ion couples in ternary and binary aqueous solutions can be applied to predict reaction ability of components in ternary and binary strong electrolyte aqueous solutions.

3) The developed universal thermodynamic model for ternary and binary aqueous solutions provides a useful method to judge structural characteristics of ternary and binary strong electrolyte aqueous solutions.

4) It is confirmed from the developed universal thermodynamic model for ternary and binary aqueous solutions that ternary and binary strong electrolyte aqueous solutions are composed of cations and anions as simple ions, H₂O as simple molecule, and other hydrous salt compounds as complex molecules.

5) The calculated mass action concentrations of structural units or ion couples in the investigated ternary and binary strong electrolyte aqueous solutions strictly follow the mass action law.

Symbol list

a_i—Reported activity of component i in aqueous solutions as , (-);

a′_i—Activity of component i in aqueous solutions defined as , (-);

b_i—Total mole number of components i before chemical reaction equilibrium in aqueous solutions, mol;

f′_i—Activity coefficient of component i relative to infinite dilute solution as standard state and molality m_i as concentration unit, (-);

I—Total ionic strength of solutes i and j in aqueous solutions as m_i+m_j, mol/kg (H₂O);

—Chemical equilibrium constant, (-);

—Transformation coefficient of component i between the calculated mass action concentration chosen pure matter as standard state and mole fraction x_i as concentration unit and the reported activity relative to infinite dilute solution as standard state and molality m_i as concentration unit, (-);

—Average of , (-);

m_i—Molality of component i in aqueous solutions, mol/kg (H₂O);

—Mole mass of H₂O, 18 g/mol;

—Total mole number of components i before chemical reaction in aqueous solutions, same as b_i, mol;

n_i—Equilibrium mole number of structural unite i or ion couple i in aqueous solutions, mol;

N_i—Mass action concentrations of structural unite i or ion couple i in aqueous solutions, (-);

—Transformed mass action concentration of structural unit i or ion couple i in aqueous solutions, (-);

R—Gas constant, 8.314?10^-3 kJ/(mol×K);

r—Ratio of molality for solute i to solute j, (-);

T—Absoulte temperature, K;

v_i—Ionic strength fraction of component i as m_i/I, (-);

x_i—Mole fraction of components i before equilibrium, (-);

—Standard molar Gibbs free energy of forming hydrous salt compound i as complex molecule, kJ/mol;

—Total mole number of all structural units in aqueous solutions under equilibrium condition, mol;

—Total mole number of structural units before dynamic equilibrium, mol.

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计算水溶液结构单元或离子对的质量作用浓度通用热力学模型及其在二元和三水溶液中的应用

杨学民¹, 赵伟洁^{2, 3}, 柴国明², 郭汉杰², 张 强³

1. 中国科学研究院 过程工程研究所 多相复杂系统国家重点实验室，北京 100190；

2. 北京科技大学 冶金与生态工程学院，北京 100083；

3. 北京中冶设备研究设计总院有限公司, 北京 100029

摘  要：基于离子与分子共存理论，建立了计算二元和三元强电解质水溶液中结构单元或离子对的质量作用浓度的通用热力学模型；同时，采用4种二元水溶液和2种三元水溶液验证该通用热力学模型。通过转换标准态和浓度单位，用所建立的通用热力学模型计算出的298.15 K时4种二元水溶液和2种三元水溶液中结构单元或离子对的质量作用浓度和文献中报道的活度值吻合得很好。因此，可采用本研究提出的通用热力学模型计算出的二元和三元水溶液结构单元或离子对的质量作用浓度预报二元和三元强电解质水溶液中组元的反应能力；同样，也可证实本研究提出的通用热力学模型的假设条件是正确和合理的，即强电解质水溶液是由阳离子和阴离子、H₂O分子和其他水合盐复杂分子组成的。基于该通用热力学模型计算出的二元和三元强电解质水溶液结构单元或离子对的质量作用浓度严格服从质量作用定律。

关键词：通用热力学模型；质量作用浓度；活度；二元水溶液；三元水溶液；离子-分子共存理论；结构单元；离子对；组元

(Edited by YANG Hua)

                     

Foundation item: Project supported by Publication Foundation of National Science and Technology Academic Books of China

Corresponding author: YANG Xue-min; Tel: +86-10-82622893; Fax: +86-10-82622893; E-mail: yangxm71@home.ipe.ac.cn

DOI: 10.1016/S1003-6326(11)60760-7