Calculating models of mass action concentrations for structural units or

ion couples in RbCl-H₂O binary system and RbCl-RbNO₃-H₂O ternary system

GUO Han-jie(郭汉杰)¹, YANG Xue-min(杨学民)², ZHAO Wei-jie(赵伟洁)¹

1. School of Metallurgical and Ecological Engineering,

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

2. State Key Laboratory of Multiphase Complex System, Institute of Process Engineering,

Chinese Academy of Sciences, Beijing 100190, China

Received 6 July 2009; accepted 4 January 2010

Abstract: Thermodynamic models of calculating mass action concentrations for structural units or ion couples in RbCl-H₂O binary and RbCl-RbNO₃-H₂O ternary strong electrolyte aqueous solutions were developed based on the ion and molecule coexistence theory at 298.15 K. A transformation coefficient is needed to compare the calculated mass action concentration and the reported activity because they are obtained at different standard states and concentration units. The results show that the transformation coefficients between the calculated mass action concentrations and the reported activities of the same structural units or ion couples in RbCl-H₂O binary and RbCl-RbNO₃-H₂O ternary strong electrolyte aqueous solutions change in a very narrow range. The transformed mass action concentrations of structural units or ion couples in RbCl-H₂O binary system are in good agreement with the reported activities. The transformed mass action concentrations of RbCl and RbNO₃ in RbCl-RbNO₃-H₂O ternary solution are also in good agreement with the reported activities, a_RbCl and , with different total ionic strengths as 0.01, 0.05, 0.1, 0.5, 1.0, 1.5, 2.0, 3.0 and 3.5 mol/kg, respectively. All those results mean the developed thermodynamic model of strong electrolyte aqueous solutions can reflect structural characteristics of RbCl-H₂O binary and RbCl-RbNO₃-H₂O ternary strong electrolyte aqueous solutions and the mass action concentration also strictly follows the mass action law.

Key words: mass action concentration; activity; ion and molecule coexistence theory; RbCl-H₂O; RbCl-RbNO₃-H₂O; structural unit; ion coupl

1 Introduction

The ion and molecule coexistence theory developed by ZHANG[1-8] since 1980s has been successfully applied to calculate mass action concentrations of structural units or ion couples in binary metallic melts[1], binary metallurgical slags[2], ternary metallic melts[3-4], ternary metallurgical slags[5-6] and multi-component complex metallurgical slags[7-8]. Expanding application scopes of the ion and molecule coexistence theory to electrolyte aqueous solutions is an interesting and challenge task. It is well-known that electrolyte aqueous solutions can be applied in many fields, such as biochemical engineering[9-12], chemical engineering [13], hydrometallurgy, environmental chemistry, and geochemistry[14]. However, activity of components in electrolyte aqueous solutions, as one of main thermodynamic properties, is not enough, or scarce and contradictory among literatures, or imprecise to fulfill its practical applications.

It has been well-known that not only ions but also some simple or complex molecules can exist in electrolyte aqueous solutions. Therefore, the ion and molecule coexistence theory, which is developed to determine mass action concentration of structural units or ion couples to present reaction ability like activity, can be applied to describe reaction ability of electrolyte aqueous solutions.

Based on the successful applications of the ion and molecule coexistence theory for calculating mass action concentrations of structural units or ion couples in KCl-H₂O, CsCl-H₂O, NaCl-H₂O, BaCl₂-H₂O[15], NaBr-H₂O, LiNO₃-H₂O, HNO₃-H₂O, KF-H₂O binary strong electrolyte aqueous solutions[16], and NaCl-KCl- H₂O[17], it can be generally deduced that the calculated mass action concentrations of structural units or ion couples have close corresponding relation with the reported activities for strong electrolyte aqueous solutions. Therefore, the calculated mass action concentrations of structural units or ion couples from the ion and molecular coexistence theory can be effectively applied to present reaction ability of components in binary and ternary strong electrolyte aqueous solutions.

To further extend application scopes of the ion and molecule coexistence theory into multi-component strong electrolyte aqueous solutions, a thermodynamic model of calculating mass action concentrations for structural units or ion couples in RbCl-RbNO₃-H₂O ternary strong electrolyte aqueous solution has been proposed according to the developed universal thermodynamic model of calculating mass action concentrations for structural units or ion couples in ternary strong electrolyte aqueous solutions[15]. To clearly describe the proposed thermodynamic model of calculating mass action concentrations for structural units or ion couples in RbCl-RbNO₃-H₂O ternary aqueous solution, a thermodynamic model of calculating mass action concentrations for structural units or ion couples in RbCl-H₂O binary strong electrolyte aqueous solutions is also proposed by applying the developed thermodynamic model of calculating mass action concentrations for structural units or ion couples in binary strong electrolyte aqueous solutions reported elsewhere[15-17].

The calculated mass action concentrations of RbCl and RbNO₃ have been compared with the reported activities[18] with different total ionic strengths as 0.01, 0.05, 0.1, 0.5, 1.0, 1.5, 2.0, 3.0 and 3.5 mol/kg in RbCl-RbNO₃-H₂O ternary solution at 298.15 K. Because the calculated mass action concentrations are usually based on pure species as standard state and mole fraction as concentration unit while the reported activity is based on infinite dilution as standard state and molality as concentration unit, a transformation coefficient should be needed for comparing the calculated mass action concentrations with the reported ones.

2 Thermodynamic model of calculating mass action concentrations

2.1 Hypotheses

The main viewpoints of the ion and molecule coexistence theory for strong electrolyte aqueous solutions can be summarized as follows.

1) Strong electrolyte aqueous solutions are composed of Na⁺, Ca²⁺, Mg²⁺, Rb⁺, Cl^-, Br^-, F^-, , etc, as simple ions, H₂O as simple molecule, and hydrous salt compounds as complex molecules. It is assumed that each cation and anion of the simple ions occupies only one position of structural units, but will take part in the reaction of forming hydro salt molecules in the form of ion couple based on electrovalence balance. Choosing RbCl as an example, RbCl can be electrolyzed or separated into two simple ions as Rb⁺ and Cl^-, respectively, as two structural units, but ions of Rb⁺ and Cl^- will take part in the reaction of forming hydro salt molecules as an ion couple in form of (Rb⁺+Cl^-) in RbCl-contained aqueous solutions if there are hydro salt molecules formed.

2) There are dynamic reaction equilibria of forming complex hydro salt molecules between ion couples and simple molecules of H₂O.

3) The structural units in strong electrolyte aqueous solutions keep continuity in the investigated concentration range of compositions.

4) Chemical reactions of forming complex hydro slat molecules in strong electrolyte aqueous solution follow the mass action law.

2.2 Calculating model of mass action concentration for structural units or ion couples in RbCl-H₂O binary system

2.2.1 Structural units and calculating model

It has been confirmed[19] that RbCl and H₂O cannot form any hydrous salt molecule at 298.15 K. Hence, the structural units in RbCl-H₂O binary solution are Rb⁺ and Cl^- as ions and H₂O as simple molecule. The molality m_i (mol/kg) is usually applied to present the concentration of components in strong electrolyte aqueous solution. Therefore, the amounts-of-substance of  RbCl and H₂O in RbCl-H₂O binary solution based on 1 kg H₂O before dynamic reaction equilibrium is expressed as,  55.6 mol, respectively. The mole fraction of RbCl and H₂O before dynamic equilibrium can be determined by

The equilibrium amount-of-substance of structural units in 1 kg H₂O in RbCl-H₂O binary solution is defined according to the ion and molecule coexistence theory as

 and =55.6

mol. The total amount of substance of all structural units in 1 kg H₂O in RbCl-H₂O binary solution under equilibrium condition, , can be calculated according to mass balance principle as

                 (1)

The mass action concentrations of each structural unit or ion couple, N_i, can be determined as

     (2a)

     (2b)

The mass balance of RbCl and H₂O in 1 kg H₂O in RbCl-H₂O binary solution can be expressed as

             (3a)

        (3b)

The following equation can be obtained from the fact that the total equilibrium molar fraction of all structural units in a system is 1.0, that is

N₁+N₃=1                                    (4)

The Eqs.(3) and (4) are the calculating thermodynamic models of mass action concentration for structural units or ion couples in RbCl-H₂O binary solution. There are three unknown parameters as N₁, N₃ and  with three independent equations in the thermodynamic model composed of Eq.(3) and Eq.(4). Therefore, the real solutions of above-mentioned mass action concentration of structural units or ion couples, N_i, and total amount of substance of all structural units, , in RbCl-H₂O binary solution can be solely solved by combining Eq.(3) and Eq.(4).

The calculated mass action concentrations of solute, i.e., N₁, is based on pure species as standard state and mole fraction, x_i, as concentration unit; however, the reported activities are usually based on infinite dilution as standard state and molality m_i as concentration unit. To compare the calculated mass action concentrations and the reported activities of solute, a transformation coefficient of RbCl,  in RbCl-H₂O binary solution is needed. The transformation coefficient,  and the transformed mass action concentration,  of RbCl in RbCl-H₂O binary solution can be described as follows:

It should be specially pointed out that although no any hydro salt molecule can be formed in RbCl-H₂O binary solution at 298.15 K, there are not independent RbCl molecules, but Rb⁺ and Cl^- ions in RbCl-H₂O binary solution. Using concept of mass action concentration for RbCl is just for convenience to compare the measured RbCl activity data with the calculated mass action concentrations of the same component because only activity data of two components in RbCl-H₂O binary solution as RbCl and H₂O can be determined and reported from viewpoints of classic experimental tests and traditional thermodynamics. The real meaning of mass action concentration for RbCl,, is the sum of mass action concentration for two structural units in RbCl as Rb⁺ and Cl^- in form of ion couple.

2.2.2 Results and discussion

The relationship between the transformed mass action concentration,, or the reported activity, a_RbCl[20], and mole fraction of RbCl, x_RbCl, in RbCl-H₂O binary solution at 298.15K is shown in Fig.1(a). Similarly, the calculated mass action concentration, , against  in RbCl-H₂O binary solution is illustrated in Fig.1(b). It can be observed from Fig.1(a) that the transformed mass action concentration,, is in good agreement with reported activity, a_RbCl[21], in the investigated RbCl concentration range. The transformation coefficients  between the transformed mass action concentrations and the reported activities of RbCl in RbCl-H₂O binary solution are also listed in Table 1 in a molality range of RbCl from 0.1 to 6.0 mol/kg(H₂O). It can be observed from Table 1 thatthe transformation coefficients of RbCl,, are more or less discrete, but an average datum of  can be calculated to be 17.95. Therefore, it can be deduced that the calculated mass action concentrations can be applied to express reaction ability of RbCl and have a close relationship with the reported activities of RbCl in RbCl-H₂O binary solution. All hypotheses used during the development of the thermodynamic model are reasonable and have got the facts of intrinsic structure of RbCl-H₂O binary solution.

Fig.1 Comparison of transformed mass action concentrationswith reported activities  of RbCl (a) and relationship between calculated mass action concentration  and mole fraction of H₂O  (b) in RbCl-H₂O binary solution at 298.15 K

Table 1 Transformation coefficients between calculated mass action concentrations and reported activities of RbCl in RbCl-H₂O solutions

2.3 Calculating model of mass action concentration for structural units or ion couples in RbCl-RbNO₃-H₂O ternary system

2.3.1 Structural units and calculating model

It has been known[19] that RbCl and RbNO₃ will be completely ionized and no hydrous salt molecules can be formed in RbCl-RbNO₃-H₂O ternary system at 298.15 K. Based on the ion and molecule coexistence theory, the structural units in RbCl-RbNO₃-H₂O ternary solution are composed of Rb⁺, Cl^- and  as simple ions, and H₂O as simple molecule. According to above-mentioned hypotheses for strong electrolyte aqueous solution, the amount-of-substance of solutes and solvent in RbCl-RbNO₃-H₂O ternary solution based on 1 kg H₂O before dynamic reaction equilibrium can be expressed asand respectively. The mole fraction of component i before equilibrium is presented as

The equilibrium amount-of-substance of above- mentioned structural units in RbCl-RbNO₃-H₂O ternarysolution can be presented as nH₂O=55.6 mol. Hence, the total amount- of- substance of all structural units in 1 kg H₂O in RbCl-RbNO₃-H₂O ternary solution under equilibrium,can be expressed according to mass balance principle as

    (6)

The mass action concentrations of structural units or ion couples in RbCl-RbNO₃-H₂O ternary solution, i.e., RbCl, RbNO₃ and H₂O, are presented as

        (7b)

          (7c)

The mass balance of solutes and solvent in 1 kg H₂O in RbCl-RbNO₃-H₂O ternary solution can be expressed, based on the ion and molecules coexistence theory, as

             (8a)

          (8b)

         (8c)

The following equation can also be obtained from the fact that the total equilibrium molar fraction of all structural units in a system is 1.0, as

N₁+N₂+N₃=1                                 (9)

The Eqs.(8) and (9) consist of the calculating thermodynamic model of mass action concentration for RbCl-RbNO₃-H₂O ternary solution. There are four unknown parameters as N₁, N₂, N₃ and  with four independent equations in the thermodynamic model composed of Eq.(8) and Eq.(9). The real solutions of above-mentioned mass action concentration of structural units or ion couples, N_i, and total amount of substance of

all structural units,in RbCl-RbNO₃-H₂O ternary

solution can be solely solved by combining Eq.(8) and Eq.(9).

To compare the calculated mass action concentrations and the reported activities of solutes, the transformation coefficients of RbCl, , and RbNO_3, , and the transformed mass action concentration,  and , can be described as follows:

,

                      (11)

2.3.2 Results and discussion

The activity coefficients of RbCl and RbNO₃ in RbCl-RbNO₃-H₂O ternary solution have been reported elsewhere[18] in a range of total ionic strength, i.e., I=+, from 0.01 to 3.5 mol/kg with differentionic strength fraction of RbNO₃,   as 0, 0.2, 0.4, 0.6, and 0.8 at each total ionic strength, respectively. The transformed mass action concentrations,  and , have been compared with the reported activities[18], a_RbCl and , with different I, i.e., various mole fraction of RbCl, x_RbCl, and RbNO₃, , as illustrated in Figs.2(a)-(r), respectively. It can be observed from Figs.2(a)-(r) that the transformed mass action concentrations,  and , are in

 good agreement of the reported activities[18],  and , in a large change range of I, i.e., with various x_RbCl and respectively. The trans- formation coefficients,  and , listed in Table 2, show that  and  keep constant with small deviations for different I, respectively. The calculated mass action concentrations of H₂O, , are also summarized in Table 2 with different I. Hence, the ion and molecule coexistence theory can be successfully applied to calculate the reaction ability of components in RbCl-RbNO₃-H₂O ternary strong electrolyte aqueous solutions.

Fig.2 Comparison of transformed mass action concentrations,and with reported activities, a_RbCl and , of RbCl and NbNO₃ in RbCl-RbNO₃-H₂O ternary solution with total ionic strength I as 0.01, 0.05, 0.1, 0.5, 1.0, 1.5, 2.0, 3.0, and 3.5 mol/kg, respectively, at 298.15 K

Table 2 Transformation coefficients, , and , in total ionic strength range from 0.01 to 3.5 mol/kg (H₂O) with different ionic strengths of RbNO₃, , as 0, 0.2, 0.4, 0.5, 0.6 and 0.8, respectively, and calculated mass action concentration of H₂O in RbCl-RbNO₃-H₂O ternary solution

The common ion in RbCl-RbNO₃-H₂O ternary system is Rb⁺, which can be generated from RbCl as well as RbNO₃. However, the calculated mass action concentration of RbCl only considers the contribution of Rb⁺ and Cl^- which exist in RbCl, but not the contribution of Rb⁺ in RbNO₃. The same method is also used to calculate the mass action concentration of RbNO₃. The treatment of the effect of common ion, i.e., Rb⁺ in RbCl-RbNO₃-H₂O ternary system, on calculation of mass action concentration for RbCl and RbNO₃ needs further investigation in viewpoint of electrochemistry for strong electrolyte aqueous solutions.

The wonderful agreement between the transformed mass action concentrations and the reported activities suggests that the developed thermodynamic model can exactly reflect the structural characteristics of RbCl-RbNO₃-H₂O ternary solution and the mass action law is applicable to RbCl-RbNO₃-H₂O ternary strong electrolyte aqueous solution.

The calculated mass action concentrations of structural units or ion couples in RbCl-RbNO₃-H₂O ternary solution are valid by transformation coefficient, which need the reported activity data based on infinite dilution as standard state and molality as concentration unit. This maybe is the main reason for more or less deviation shown in Figs.2(a)-(r). The reasonable method is to compare the calculated mass action concentration of components and the measured activity with pure matter as standard state and mole fraction as concentration unit.

3 Conclusions

1) The calculated mass action concentrations of structural units or ion couples in RbCl-H₂O binary and RbCl-RbNO₃-H₂O ternary strong electrolyte aqueous solutions are in good agreement with the reported activities from literatures. Therefore, the developed thermodynamic model of calculating mass action concentrations for structural units or ion couples in RbCl-H₂O binary and RbCl-RbNO₃-H₂O ternary strong electrolyte aqueous solutions can be successfully applied to predict the reaction ability of components.

2) The ion and molecule coexistence theory in combination with the mass action law (if necessary when hydrous salt molecules as complex molecules exist) can be successfully applied to describe the structural characteristics of RbCl-H₂O binary and RbCl-RbNO₃- H₂O ternary strong electrolyte aqueous solutions.

3) As there are visible deviations between the calculated mass action concentrations and the reported activities in high concentration range for two investigated aqueous solutions, it is absolutely necessary to carry out further investigation on reaction ability of structural units or ion couples in strong electrolyte aqueous solutions based on pure species as standard state and mole fraction as concentration unit.

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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(09)60265-X

 (Edited by YANG Bing)