Thermodynamic Modeling of Surface Tension of Aqueous Electrolyte Solution by Competitive Adsorption Model

For studying the aqueous electrolyte solution in porous media, distillation, extraction process and liquid-liquid dispersion, the calculation of surface tension of aqueous solutions is required [1]. So, different equations have been developed to calculate surface tension of mineral salts. Ariyama [2], Lorenz [3], Young and Grinstead [4], and Glejm and Shelomov [5] formulated useful equations as group contribution method for calculation of surface tension of some limited binary electrolyte-water systems. Oka [6] proposed an equation for calculation of surface tension based on the concentration of solution, electronic charge, dielectric constant of water, ionic charge and Avogadro’s constant. Later Hovarth [7] developed this equation by introducing the ionic strength and degree of dissociation into the Oka’s model [6]. Onsager and Samaras [8], obtained a relation based on the temperature, dielectric constant of water and concentration of solution for calculation of surface tension of electrolyte solution. Schmutzer [9],considered the osmotic coefficient as an important factor for calculation of surface tension of electrolyte solution. Adding a proportional factor of anion concentration to the surface tension of water, the surface tension of aqueous electrolyte solution was determined by Abramazon and Gaukhberg [10]. This parameter was considered as a function of the inverse of square of ionic radius and anion charge. Li et al. [1] developed a new model for calculation of surface tension of single and mixed electrolyte solution. In this model, the surface layer is considered as a distinct phase where the electrolytes could be entered into it from another phases. The surface tension was obtained using the proportion of molality of salt in surface layer to liquid bulk phase. While this model had satisfactory results in low concentration of electrolytes, in high concentration the calculated surface tensions were not in good agreement with experimental data. Yu et al. [11] combined Li et al. [1] model with modified mean spherical approximation (MSA) as osmotic coefficient model. The results showed that the calculated surface tensions in highly concentrated regions were improved. Furthermore, the Langmuir gas-solid adsorption model was used at equilibrium condition for calculation of surface tension of mineral salts by Li and Lu [12]. The results indicated the satisfactory agreement with experimental data. Sadeghiet al. [13] used the combination of MSA model [14] with the Ghotbi and Vera [15] and the Mansoori et al. [16] equations of state, for correlation of the surface tension of single aqueous solution. Also the surface tensions of different mixed aqueous solutions were predicted by this approach. The results indicate the satisfactory agreement between calculated and experimental data [13]. In this paper, a new model for calculation of surface tension of the electrolyte systems is developed using the Langmuir adsorption equation and E-NRTL [17] model. The adjustable parameters of this model are obtained by experimental data of surface tension at single temperature. Then the model is verified by prediction of surface tension of 65 binary electrolyte-water systems and 17 ternary electrolyte systems.


Introduction
For studying the aqueous electrolyte solution in porous media, distillation, extraction process, and liquid-liquid dispersion, the calculation of surface tension of aqueous solutions is required [1]. So different equations have been developed to calculate surface tension of mineral salts. Ariyama [2], Lorenz [3], Young and Grinstead [4], and Gleim and Shelomov [5] formulated useful equations as group contribution method for calculation of surface tension of some limited binary electrolyte-water systems. Oka [6] proposed an equation for calculation of surface tension based on the concentration of solution, electronic charge, dielectric constant of water, ionic charge, and Avogadro's constant. Later Hovarth [7] developed this equation by introducing the ionic strength and degree of dissociation into Oka's model [6]. Onsager and Samaras [8] obtained a relation based on the temperature, dielectric constant of water, and concentration of solution for calculation of surface tension of electrolyte solution. Schmutzer [9] considered the osmotic coefficient as an important factor for calculation of surface tension of electrolyte solution. Adding a proportional factor of anion concentration to the surface tension of water, the surface tension of aqueous electrolyte solution was determined by Abramazon and Gaukhberg [10]. This parameter was considered as a function of the inverse of square of ionic radius and anion charge. Li et al. [1] developed a new model for calculation of surface tension of single and mixed electrolyte solution. In this model, the surface layer is considered as a distinct phase where the electrolytes could be entered into it from other phases. The surface tension was obtained using the proportion of molality of salt in surface layer to liquid bulk phase. While this model had satisfactory results in low concentration of electrolytes, in high concentration the calculated surface tension was not in good agreement with experimental data. Yu et al. [11] combined Li et al. [1] model with modified mean spherical approximation (MSA) as osmotic coefficient model. The results showed that the calculated surface tension in highly concentrated regions was improved. Furthermore, the Langmuir gas-solid adsorption model was used at equilibrium condition for calculation of surface tension of mineral salts by Li and Lu [12]. The results indicated the satisfactory agreement with experimental data. Sadeghi et al. [13] used the combination of MSA model [14] with the Ghotbi and Vera [15] and the Mansoori et al. [16] equations of state, for correlation of the surface tension of single aqueous solution. Also the surface tension of different mixed aqueous solutions was predicted by this approach. The  results indicate the satisfactory agreement between calculated and experimental data [13].
In this paper, a new model for calculation of surface tension of the electrolyte systems is developed using the Langmuir adsorption equation and E-NRTL [17] model. The adjustable parameters of this model are obtained by experimental data of surface tension at single temperature. Then the model is verified by prediction of surface tension of 65 binary electrolyte-water systems and 17 ternary electrolyte systems.

Thermodynamic Modeling
For calculation of surface tension of electrolyte system, the aqueous electrolyte solution-vapor system is supposed as three different phases: bulk vapor phase, surface phase, and bulk liquid phase ( Figure 1). The surface phase is considered as distinct layer for adsorption of electrolyte from liquid phase. In this system, the chemical potential of water in liquid bulk phase and surface would be defined as follows [1]: where , , , and sol represent the chemical potential, activity, partial molar area, and surface tension of solution. The subscripts , , , 0 , and 0 refer to water, liquid phase, surface phase, reference state of liquid phase, and reference state of surface phase, respectively. At equilibrium condition, the chemical potential of water in surface and liquid phase is equal. So we have Rewriting the above equation for pure water, it yields the following equation: It would be worth noting that the activity of pure water is unity and the partial molar area for pure water is equal to the molar area.
Substituting the above equation into (2) results in the following equation: It is assumed that in the above equation the partial molar area and molar area of water are equal. The molar area of water is calculated by [18] where and are molar volume of water and Avogadro number.
Using the osmotic definition ( ) instead of activity of water, (4) converts to the following equation: where is molality of electrolyte in aqueous solution and V is stoichiometric coefficient. For calculation of the molality of electrolyte in surface phase ( ), the fraction of adsorbed electrolyte ( ) on the interface between vapor and liquid phases could be related to the excess area (Γ) as [12] = Γ Γ 0 , where superscript 0 means saturated condition. On the other hand, the excess area is defined as moles of electrolyte in surface per area of surface [19]. So, where is mole number and superscript indicates the surface phase. The saturated molality of surface phase is considered as fraction of molality of liquid bulk phase or where is liquid bulk phase and 0 is a constant. Using the Langmuir gas-solid adsorption [20] and equality of adsorption and desorption rate for an electrolyte in surface phase [21], the adsorbed fraction of electrolyte on the surface phase is obtained as [12] = 1 + .
Combining (6)-(8), the molality of electrolyte in surface phase would be obtained as where * = 0 . So, substituting (11) into (6), the surface tension of pure aqueous electrolyte solution is calculated. It is mentioned that the two adjustable parameters * and are obtained from Journal of Thermodynamics 3 fitting the experimental surface tension data to the calculated values (see (6)) at single temperature.
For mixed electrolyte solution the surface tension would be calculated by the following equation: Assuming competitive adsorption between electrolytes in surface phase [12], the molality of surface phase is

Result and Discussion
For studying the surface tension of electrolyte solution, 65 binary electrolyte systems and 17 ternary electrolyte systems are selected. The surface tension of these electrolyte solutions is obtained by competitive adsorption model (see (6) or (12)). In this model, for calculation of osmotic and activity coefficients, E-NRTL [17] model is used. Using the regression of model with experimental surface tension of binary electrolyte-water system, the adjustable parameters of competitive adsorption model ( * , ) are optimized. The objective function is defined as follows: where AAD, , and superscripts cal and exp are average absolute deviation, number of data points, and calculated and experimental data of surface tension, respectively. The results for different binary electrolyte system are given in Table 1. So, at single temperature for each salt which is given in Table 1, the surface tension is correlated.
As it is shown in Table 1, when the first adjustable parameter ( * ) tends to zero or small positive value or the second adjustable parameter ( ) becomes a large number, the molality of electrolyte system in surface phase is close to zero. Moreover, when * is lower than unity and is not moderately large number, the molality of electrolyte in surface phase decreases with respect to the liquid bulk phase. In these cases, which point out to the mineral salts, the rate of adsorption in surface layer would be lower than desorption rate and surface tension of electrolyte solution is increased with respect to the surface tension of water. Strong inorganic acids, organic acetates, and propionates which have high vapor pressure tend to escape from liquid phase to surface phase and consequently the concentration of these components in surface phase would be greater than liquid phase [11]. For these electrolytes, the first adjustable parameter is higher than unity and so the molality of electrolyte in surface phase becomes greater than the liquid bulk phase and consequently the surface tension of aqueous electrolyte solution increases with respect to the liquid bulk phase.
For other temperatures, the value of surface tension of electrolyte aqueous solution is predicted and the results are  given in Table 2 Table 3. The overall AAD for prediction of surface tension of mixed electrolyte solutions is 1.7%.       of the competitive adsorption model could introduce this new model as effective one.