Isobaric vapor-liquid equilibria of ternary lead–tin–antimony alloy system at 2 Pa

In this study, experimental vapor–liquid equilibria data of the ternary Pb–Sn–Sb alloy system are determined using a new experimental method. The experimental VLE data passed the thermodynamic consistency test (Van Ness test), suggesting that the experimental results are reliable. The activities of the components of Pb–Sn–Sb ternary alloy and those of the corresponding three constituent binaries were calculated using the Wilson equation. The predicted values are in good agreement with the data determined from experiments, and the average relative deviation and average standard deviation were smaller than ± 4.00% and ± 0.03 for all constituent binaries, respectively, which indicates that the Wilson equation is reliable for calculating the activity of the components of the Pb–Sn–Sb ternary alloy. The VLE data of the Pb–Sn and Sb–Sn binary alloys and Pb–Sn–Sb ternary alloy were calculated based on the VLE theory and Wilson equation. The calculated VLE data were in good agreement with the data determined from experiments, indicating that this method is reliable for calculating the VLE of alloy systems. The proposed study offers a valid method for analyzing the composition of products that are dependent on the distillation temperature and system pressure during vacuum distillation, which is of great significance to the experimental design of this process.


Introduction
A significant amount of Pb-Sn-Sb alloys is produced from global smelters annually due to the similar behavior of Pb, Sn, and Sb during the smelting process. This is because of their similar physicochemical properties and association in minerals. Pb, Sn, and Sb are mainly used for the manufacturing of lead-acid batteries, solders, and bearing alloys, which generates large amounts of waste Pb-Sn-Sb alloys. 3 were stored in a container filled with high-purity Ar gas to prevent oxidation. Ar gas (mass fraction purity ≥ 0.99999) was obtained from Kunming Messer Gas Products Co., Ltd., China. All chemicals were used without further purification. The summary of CAS#, suppliers, and purities of chemicals used in this study are listed in Table 1.

.1 Alloy preparation
Pb-Sn-Sb alloys were prepared using a quartz tube equipped with a vacuum pump (2XZ-4Z, China), Ar gas cylinder, and resistance furnace. Temperatures were measured using a thermocouple (type K) obtained from Jinhong Instrument Co., Ltd., China with an uncertainty of ± 1 K. Pressures were monitored with a combined pressure and vacuum gauge with an uncertainty of ± 2 Pa. An OHAUS (Pine Brook, NJ USA) balance, Model AR 1140/C, with an uncertainty of ± 0.0001 g was used for weighing.
The three high-purity metals were accurately weighed and melted into an alloy with a predetermined composition (xPb = 0.2000, xSn/xSb = 1/3). Fig. 1 shows the schematic diagram of the experimental device used for the preparation of the Pb-Sn-Sb alloys. For each run, 44.9659 g of Pb particles, 25.7621 g of Sn particles, and 79.2720 g of Sb polycrystalline lumps were put in a high-purity graphite crucible (inner diameter 32 mm × Ht 110 mm, purity 99.99%). The graphite crucible was loaded in a quartz tube (inner diameter 82 mm × Ht 350 mm), and the quartz tube was sealed with a silicone plug with three holes (inner diameter 6 mm × Ht 60 mm), which were used for setting the thermocouple, gas inlet, and gas outlet. Then, the quartz tube was placed in a crucible resistance furnace, as shown in Fig. 1. Thereafter, the system was evacuated to approximately 3 Pa and then filled with high-purity Ar gas to prevent oxidation and heated to and kept at 904 K for 30 min to melt the Pb, Sn, and Sb into a Pb-Sn-Sb alloy under flowing high-purity Ar gas (10 ml·min −1 ). The alloy was remelted three or four times as required to ensure its homogeneity. The resistance furnace was cooled to approximately 30 o C before the alloy was removed from the graphite crucible. The mass losses of the alloy were smaller than 0.01%; therefore, the weighed compositions of the alloys were considered to be correct.

VLE measurements
The VLE data of the Pb-Sn-Sb alloy were determined using a steel chamber, as described in our previous work [5].
In each run, approximately 150 g of the Pb-Sn-Sb alloy (xPb = 0.2000, xSn/xSb = 1/3) was placed into a stainless steel chamber. The chamber, in which a capillary had been previously inserted, was welded and evacuated to 2 Pa. The chamber was then placed in a resistance furnace [5]. The experiments were conducted at 1023, 1073, 1123, 1173, and 1223 K, respectively, until the system reached equilibrium.
When the composition of the vapor phase and liquid phase no longer changed, the holding time was considered as the equilibrium time. At least four experiments were conducted at each temperature with a time interval of 30 min to obtain the equilibrium time. The composition of the vapor phase and liquid phase began to reach a constant value after over 3 h under the conditions of T = 1023 K and P = 2 Pa. Although the time required for the system to achieve the equilibrium decreases gradually with an increase in the distillation temperature, the final distillation time was 4 h for all the experiments in this study to ensure that the system sufficiently approaches the phase equilibrium.
At the end of each experiment, the steel chamber was quickly removed from the furnace and quenched in water. Therefore, the volatiles cooled on the sidewall of the chamber, whereas the residues cooled at the bottom of the chamber.
Finally, the volatiles and residues were collected from the sidewall and bottom of the chamber, respectively. The Sb and Pb content in the vapor phase (volatiles) and liquid phase (residues) were determined by titration analysis, and the detailed determination procedure is available in Ref. [5]. The Sn content in the volatiles and residues were obtained by solving the equation of mass balance accounting for the VLE. Using the method in Ref. [5], the combined standard uncertainty at the experimental conditions in this study was estimated to be ± 0.003 in mole fraction. 5

Applied thermodynamic model
The activities or activity coefficients of the alloys in the liquid phase are necessary for calculating the VLE data. Thermodynamic calculation is an effective method for this application, and the Wilson equation is considered as one of the most successful models [10,11]. The excess free energy G E [12] of a multicomponent system can be derived from the Wilson equation and expressed as follows: where R is the gas constant (8.314 J‧K -1 ‧mol -1 ); xi and xj are the molar fraction of component i and j, respectively; T is the absolute temperature; and the adjustable parameters Aij and Aji are expressed as: where gii, gij, and gjj are the i-i, i-j, and j-j interaction energies, respectively (gij=gji); Vi and Vj are the molar volumes of component i and j, respectively, as shown in Table  2 [13]. Table 2 The molar volumes of the components of Pb-Sn-Sb ternary alloy [13]. For a binary mixture i-j, the activities of i and j components can be calculated from Eq. (1), respectively, which can be expressed as follows: When xi or xj infinitely approaching zero, the activity coefficients of i and j (i.e., γi ∞ and γj ∞ ) can be obtained from Eqs. (3) and (4), respectively, which can be expressed as: The initial values of Aij and Aji can be obtained from the given data of γi ∞ and γj ∞ [14] by computing repeatedly (n+1) times until calculation process is available in Ref. [15]. Assuming that − (gij − gjj)/R and − (gji − gii)/R are independent of temperature, and the values of Aij and Aji at a certain temperature (T1) are known, the values of Aij and 6 Aji at the required temperature (T2) can be calculated from Eq. (2).
The required parameters Aij and Aji of the constituent binaries of the Pb-Sn-Sb ternary system are shown in Table 3. Table 3 The values of Aij and Aji, γi ∞ and γj ∞ [14] for the constituent binaries of the Pb-Sn-Sb ternary system. Letting the Pb-Sn-Sb ternary system be a 1-2-3 system, the activity coefficient of component 1 of the system can be expressed as follows:

Saturation vapor pressure
The vapor pressure of pure substances in the saturated state is also indispensable for the calculation of VLE. It can be calculated using the following vapor pressure equation [16]: where P s is the saturation vapor pressure of pure substances in Pa; A, B, C, and D [16] are the component specific coefficients of vapor pressure; T is the temperature in K. The saturation vapor pressure equations for Sn, Pb, and Sb are listed in Table 4. Table 4 The saturation vapor pressure equation for Sn, Pb and Sb [16]. Substance

VLE Calculation
The thermodynamic condition for the VLE is the equality of the fugacities of each component in each phase, which is expressed as follows [17]: When the fugacity coefficient is used in Eq. (9), we obtain: Based on Eqs. (9) and (10), the vapor-phase composition yi can be expressed as follows [17]:

Results and Discussion
In this section, we present and discuss the VLE data of the ternary Pb-Sn-Sb system. After presenting the experimental VLE data of the ternary system, the calculated results are shown. The calculated results and experimental data were compared for validation purposes.

Experimental results
The experimental VLE data of the Pb-Sn-Sb alloy obtained from the new method are presented in Table 5. Table 5 shows that the Sb content in the residue (liquid phase) decreases with an increase of temperature. At 1123 K, the Sb content in the residue was 0.5726 (mole fraction), and the Pb and Sn content were increased to 0.2211 and 0.2063, respectively. A small quantity of Pb and Sn will evaporate into the volatiles (vapor phase) after the temperature was increased; therefore, the Pb and Sn content in the volatiles, i.e., yPb and ySn, increased to 0.03945 and 0.0024, respectively. However, the Sb content in the volatiles was much higher than the Sn and Pb content therein, indicating that the vapor phase mainly comprised Sb. This demonstrates that it is effective to separate Sb from a Pb-Sn-Sb alloy by vacuum distillation. Table 5 Experimental VLE data for temperature T, liquid-phase mole fraction xi, and 8 vapor-phase mole fraction yi, for the Pb (1) + Sn (2) + Sb (3) ternary system, measured at 2.0 Pa a a u (T) = ± 2 K, u (x) = u (y) = ± 0.003.

Thermodynamic consistency test
The thermodynamic consistency test is a criterion for determining the correctness of the experimental VLE data. In this study, the Van Ness test [19][20] was employed to test the thermodynamic consistency of the VLE data for Pb-Sn-Sb alloy, which can be expressed as follows: where y(MAD) is the absolute deviation, yi exp is the experimental data, yi cal is the calculated results from the Wilson equation, and N is the number of experimental points.
According to the definition of the Van Ness test, the experimental VLE data are reliable if y(MAD) is smaller than 1. Table 6 shows that the value of y(MAD) for Pb, Sn, and Sb is 0.8134, 0.2850, and 0.7300, respectively, indicating that the experimental VLE data obtained by the new method are correct and reliable. Table 6 Thermodynamic consistency test result of Pb-Sn-Sb alloy system at 2.0 Pa.

Activity coefficients
The activities of the components of the Pb-Sn binary alloy at 1050 K and Pb-Sb and Sn-Sb binary alloys at 905 K were calculated using Eqs. (3) and (4), respectively, as shown in Fig. 2. The calculated values of activities are in good agreement with the experimental data. To quantify the deviations of the model predictions from the data determined from experiments, the average relative deviation (Si) and average standard deviation were calculated (Si * ) using Eqs. (16) and (17), respectively, as shown in Fig.  2.
where n is the number of data points; ai,exp is the experimental data of activity, and ai,cal is the calculated value from the Wilson equation.   Figure 2 shows that the average relative deviations of aPb and aSn were ± 2.870% and ± 3.410%, respectively, and the average standard deviations of aPb and aSn were ± 0.024 and ± 0.021, respectively. In addition, the average relative deviation and average standard deviation for all constituent binaries was smaller than ± 4.00% and ± 0.03, respectively, which demonstrates that the Wilson equation is effective for calculating the activities of the components of the constituent binaries of the Pb-Sn-Sb alloy system.
For the Pb-Sn-Sb ternary alloy, the activity of each component can be calculated from Eq. (7), in which only the simple parameters of the corresponding constituent binary alloys in needed. The activities of the components of the Pb-Sn-Sb ternary alloy were obtained by substituting the Wilson parameters for the Pb-Sn, Sb-Sn, and Pb-Sb binary systems into Eq. (7), as shown in Table 7. As the reliable experimental data of the activity of the Pb-Sn-Sb ternary system were not available, the reliability of calculated activity of the ternary system could not be tested. Nevertheless, according to the results of the binary system, the prediction accuracy of the Wilson equation is high. Furthermore, a good application effect of the Wilson equation has been demonstrated for predicting the thermodynamic properties of ternary Pb-based alloys [15].

VLE of Pb-Sn-Sb alloy system
The VLE was calculated using an iterative process of estimating a temperature and calculating the partial pressure of the components from the activity coefficients and the vapor pressures of the pure liquids at a given temperature, until the sum of the partial pressures P equals the total pressure initially set. The steps required to calculate the VLE are as follows: 1. Calculate the saturation temperature Ti,b for component i from Eq. (8).
2. Set a series of xi and calculate the approximate temperature T from  (4) or Eq. (7). 5. Calculate Ki and yi from Eq. (12). 6. Substitute yi into Eq. (14). If the absolute difference is less than a tolerance value, then output Tb, xi, and yi; otherwise, estimate a new value for T and return to step 3 until the sum of the partial pressures P equals the total pressure initially set (e.g., 2 Pa). Fig. 3 shows the flowchart for calculating the VLE of the Pb-Sn-Sb alloy system. Fig. 3. Flowchart for calculating the VLE of the Pb-Sn-Sb ternary alloy. In our previous work, the VLE of Pb-Sn and Sb-Sn alloy systems were experimentally investigated and calculated using the molecular interaction volume 12 model [5]. In this study, the VLE was calculated using the Wilson equation for Pb-Sn and Sb-Sn system at 2 Pa and the experimental data are shown in Fig. 4.  Fig. 4 shows that Pb and Sb can be concentrated in the vapor phase, and Sn can be concentrated and purified in the liquid phase by vacuum evaporation. The Sn content in the Pb or Sb was low enough for them to be considered as refined Pb and refined Sb. Fig. 4 suggests that the calculated values are more reliable than the experimental data for the liquid phase. This may be due to the following reasons. (1) The process did not sufficiently reach the thermodynamic equilibrium, and the determination of the equilibrium state could have had a large error. In other words, the Pb or Sb in the melt was not fully volatilized and remained in the liquid phase, resulting in a higher Pb and Sb content therein. (2) The system pressure was not constant and could have been larger than 2 Pa.
As the experimental VLE data of the Pb-Sb alloy system determined using new method were not available in the literatures until now, only the activities of the components were predicted using the Wilson equation (Fig. 2). It can be seen from the calculations that the predicted value of the Wilson equation is very consistent with the data determined from experiments.
Following steps 1-6 as described previously, the VLE data of the Pb-Sn-Sb ternary system were obtained. The experimental VLE data and calculated vapor-phase compositions from the Wilson equation at 2 Pa are listed in Table 5. The calculated results from the Wilson equation agree well with the data determined from experiments. Meanwhile, under the condition of 2 Pa, the deviations between the calculated values from the Wilson equation and the experimental data in the vapor phase are shown in Fig. 5. The VLE data of the Pb-Sn-Sb ternary alloy calculated by this method are also in agreement with the data determined from experiments, although some deviations are observed especially at high temperatures.  The deviations mainly come from the following aspects: (1) The activity coefficient predictions of the constituent binary systems have an impact on the multicomponent systems, especially for asymmetric systems. (2) For multicomponent systems, only binary interactions between atoms are considered, and when Wilson equation is extended to multicomponent alloy systems, atomic interactions between the constituents are usually ignored. (3) The assumptions that were adopted in the process of VLE calculation resulted in certain errors. (4) There was large error in the calculated saturated vapor pressure of the pure components of the alloy. (5) The new experimental conditions are still different from the ideal equilibrium state. So far, the above-mentioned factors are still inevitable.

Conclusions
Reliable VLE data of the Pb-Sn-Sb ternary alloy system were determined at 2 Pa using a new experimental method. The experimental results indicate that Sb can be satisfactorily removed from Pb-Sn-Sb alloy melt. The VLE data of the Pb-Sn, Sb-Sn, and Pb-Sn-Sb alloy systems were predicted based on Wilson equation and the VLE theory. The good agreement between the experimental data and calculated values of VLE demonstrates that this calculation method is reliable for alloy systems, and it can serve as a convenient and effective method for calculating their VLE. Based on the VLE data, separation conditions can be conveniently chosen and product compositions can be quantitatively predicted. Fig. 1. Schematic of the experimental apparatus for the preparation of Pb-Sn-Sb alloys. Fig. 2. Comparison of the experimental data (symbols) [14] of activities with the calculated values from the Wilson equation (solid lines): (a) Pb-Sn alloy system at 1050 K; (b) Pb-Sb alloy system at 905 K; and (c) Sn-Sb alloy system at 905 K. Fig. 3. Flowchart for calculating the VLE of the Pb-Sn-Sb ternary alloy.   Table 2 The molar volumes of the components of Pb-Sn-Sb ternary alloy [13]. Table 3 The values of γi ∞ , γj ∞ [14], Aij and Aji for the constituent binaries of the Pb-Sn-Sb ternary system. Table 4 The saturation vapor pressure equation for Sn, Pb and Sb [16]. Table 5 Experimental VLE data for temperature T, liquid-phase mole fraction xi, and vapor-phase mole fraction yi, for the Pb (1) + Sn (2) + Sb (3) ternary system, measured at 2.0 Pa. Table 6 Thermodynamic consistency test result of Pb-Sn-Sb alloy system at 2.0 Pa Table 7 The calculated activities of components of Pb-Sn-Sb alloy with Wilson equation in the temperature range of 1073-1273 K.