Calculation of vapor-liquid equilibrium of binary precious metal alloys using modified molecular interaction volume model

MIVM was obtained from statistical thermodynamics and the basic characteristics of the nonrandom transfer of liquid molecules [18]. The expression of mole excess Gibbs free energy G_{E m} of MIVM is:

$$\begin{eqnarray}&&\displaystyle \frac{{G}_{m}^{E}}{RT}=\displaystyle \sum _{i=1}^{C}{x}_{i}\,\mathrm{ln}\,\displaystyle \frac{{V}_{mi}}{\displaystyle \sum _{j=1}^{C}{x}_{j}{V}_{mj}{B}_{ji}}-\displaystyle \frac{1}{2}\displaystyle \sum _{i=1}^{C}{Z}_{i}{x}_{i}\left(\displaystyle \frac{\displaystyle \sum _{j=1}^{C}{x}_{j}{B}_{ji}\,\mathrm{ln}\,{B}_{ji}}{\displaystyle \sum _{j=1}^{C}{x}_{j}{B}_{ji}}\right)\end{eqnarray} \tag{ 1 }$$

For a binary system i-j, equation (1) can be rewritten as:

$$\begin{eqnarray}\begin{array}{lll}\displaystyle \frac{{G}_{m}^{E}}{RT} & = & {x}_{i}\,\mathrm{ln}\,\displaystyle \frac{{V}_{mi}}{{x}_{i}{V}_{mi}+{x}_{j}{V}_{mj}{B}_{ji}}+{x}_{j}\,\mathrm{ln}\,\displaystyle \frac{{V}_{mj}}{{x}_{j}{V}_{mj}+{x}_{i}{V}_{mi}{B}_{ij}}\\ & & -\,\displaystyle \frac{{x}_{i}{x}_{j}}{2}\left(\displaystyle \frac{{Z}_{i}{B}_{ji}\,\mathrm{ln}\,{B}_{ji}}{{x}_{i}+{x}_{j}{B}_{ji}}+\displaystyle \frac{{Z}_{j}{B}_{ij}\,\mathrm{ln}\,{B}_{ij}}{{x}_{j}+{x}_{i}{B}_{ij}}\right)\end{array}\end{eqnarray} \tag{ 2 }$$

According to equation (2) and (∂G_{E m} /∂x_i )_{T, P, xi≠j}  = RT ln γ_i , the activity coefficients of i and j are:

$$\begin{eqnarray}&&\begin{array}{l}\mathrm{ln}\,{\gamma }_{i}=1+\,\mathrm{ln}\,\displaystyle \frac{{V}_{mi}}{{x}_{i}{V}_{mi}+{x}_{j}{V}_{mj}{B}_{ji}}-\displaystyle \frac{{x}_{i}{V}_{mi}}{{x}_{i}{V}_{mi}+{x}_{j}{V}_{mj}{B}_{ji}}-\displaystyle \frac{{x}_{j}{V}_{mi}{B}_{ij}}{{x}_{j}{V}_{mj}+{x}_{i}{V}_{mi}{B}_{ij}}\\ \,\,\,\,\,-\displaystyle \frac{{{x}_{j}}^{2}}{2}\left[\displaystyle \frac{{Z}_{i}{{B}_{ji}}^{2}\,\mathrm{ln}\,{B}_{ji}}{{\left({x}_{i}+{x}_{j}{B}_{ji}\right)}^{2}}+\displaystyle \frac{{Z}_{j}{B}_{ij}\,\mathrm{ln}\,{B}_{ij}}{{\left({x}_{j}+{x}_{i}{B}_{ij}\right)}^{2}}\right]\end{array}\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&\begin{array}{l}\mathrm{ln}\,{\gamma }_{j}=1+\,\mathrm{ln}\,\displaystyle \frac{{V}_{mj}}{{x}_{j}{V}_{mj}+{x}_{i}{V}_{mi}{B}_{ij}}-\displaystyle \frac{{x}_{i}{V}_{mj}{B}_{ji}}{{x}_{i}{V}_{mi}+{x}_{j}{V}_{mj}{B}_{ji}}-\displaystyle \frac{{x}_{j}{V}_{mj}}{{x}_{j}{V}_{mj}+{x}_{i}{V}_{mi}{B}_{ij}}\\ \,\,\,\,\,\,-\displaystyle \frac{{{x}_{i}}^{2}}{2}\left[\displaystyle \frac{{Z}_{j}{{B}_{ij}}^{2}\,\mathrm{ln}\,{B}_{ij}}{{\left({x}_{j}+{x}_{i}{B}_{ij}\right)}^{2}}+\displaystyle \frac{{Z}_{i}{B}_{ji}\,\mathrm{ln}\,{B}_{ji}}{{\left({x}_{i}+{x}_{j}{B}_{ji}\right)}^{2}}\right]\end{array}\end{eqnarray} \tag{ 4 }$$

where x_i and x_j are the molar fractions; V_mi and V_mj are the molar volumes; Z_i and Z_j are the first coordination numbers; B_ij and B_ji are the pair-potential energy interaction parameter which are defined as equation (5). Z_i can be calculated from equation (6).

$$\begin{eqnarray}&&{B}_{ij}=\exp \left(-\displaystyle \frac{{\varepsilon }_{ij}-{\varepsilon }_{jj}}{kT}\right)\,{B}_{ji}=\exp \left(-\displaystyle \frac{{\varepsilon }_{ji}-{\varepsilon }_{ii}}{kT}\right)\end{eqnarray} \tag{ 5 }$$

Where ε_ii , ε_jj and ε_ij (ε_ij = ε_ji ) are the pair-potential energy of i-i, j-j and i-j, respectively.

$$\begin{eqnarray}&&{Z}_{i}=\displaystyle \frac{4\sqrt{2\pi }}{3}\left(\displaystyle \frac{{r}_{mi}^{3}-{r}_{0i}^{3}}{{r}_{mi}-{r}_{0i}}\right)\,\cdot \,{\rho }_{i}\,\cdot \,{r}_{mi}\,\cdot \,\exp \left[\displaystyle \frac{{\rm{\Delta }}{H}_{mi}({T}_{mi}-T)}{{Z}_{c}RT{T}_{mi}}\right]\end{eqnarray} \tag{ 6 }$$

where r_mi is equal to the atomic diameter, r_mi = σ_i ; r_0i (r_0i = 0.918d_{cov, i} ) is proportional to the atomic covalent diameter d_{cov, i} ; ρ_i is the molecular number density; ΔH_mi and T_mi are the melting enthalpy and melting temperature, respectively; Z_c (Z_c  = 12) is a close-packed coordination.

M-MIVM and MIVM are deduced from the same partition function [16]. However, M-MIVM is obtained from different hypothesis and theories. In the derivation of M-MIVM, the Scatchard-Hildebrand theory [19, 20] and radial distribution function [19] are further introduced on the base of cell theory [18, 21] which are employed by MIVM. The mole excess Gibbs free energy G_{E m} of M-MIVM is:

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{{G}_{m}^{E}}{RT}=-\displaystyle \sum _{i=1}^{C}{x}_{i}\,\mathrm{ln}\left(\displaystyle \sum _{j=1}^{C}{x}_{j}\displaystyle \frac{{V}_{mj}}{{V}_{mi}}{B}_{ji}\right)+\\ \,-\,\displaystyle \sum _{j\gt 1}^{C}\displaystyle \sum _{i=1}^{C-1}{x}_{i}{x}_{j}\left(\displaystyle \frac{{A}_{ji}}{\displaystyle \sum _{l=1}^{C}{x}_{l}\displaystyle \frac{{V}_{ml}}{{V}_{mi}}{B}_{li}}+\displaystyle \frac{{A}_{ij}}{\displaystyle \sum _{l=1}^{C}{x}_{l}\displaystyle \frac{{V}_{ml}}{{V}_{mj}}{B}_{lj}}\right)\end{array}\end{eqnarray} \tag{ 7 }$$

For i-j binary system, equation (7) are rewritten as:

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{{G}_{m}^{E}}{RT}=-{x}_{i}\,\mathrm{ln}\left({x}_{i}+{x}_{j}\displaystyle \frac{{V}_{mj}}{{V}_{mi}}{B}_{ji}\right)-{x}_{j}\,\mathrm{ln}\left({x}_{j}+{x}_{j}\displaystyle \frac{{V}_{mi}}{{V}_{mj}}{B}_{ij}\right)\\ \,\,\,+{x}_{i}{x}_{j}\left(\displaystyle \frac{{A}_{ji}}{{x}_{i}+{x}_{j}\tfrac{{V}_{mj}}{{V}_{mi}}{B}_{ji}}+\displaystyle \frac{{A}_{ij}}{{x}_{j}+{x}_{j}\tfrac{{V}_{mi}}{{V}_{mj}}{B}_{ij}}\right)\end{array}\end{eqnarray} \tag{ 8 }$$

where V_mi and V_mj are liquid molar volumes of i and j, respectively; the volume parameters (B_ij and B_ji ) and the energy parameters (A_ij and A_ji ) are defended as equation (9) and equation (10) as follows, respectively.

$$\begin{eqnarray}&&{A}_{ij}=-\displaystyle \frac{3K}{{k}_{B}}{C}_{j}^{0}\left({\varepsilon }_{ij}-{\varepsilon }_{jj}\right)\,{A}_{ji}=-\displaystyle \frac{3K}{{k}_{B}}{C}_{i}^{0}\left({\varepsilon }_{ji}-{\varepsilon }_{ii}\right)\end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray}&&{B}_{ij}={p}_{ij}/{p}_{jj}\,{B}_{ji}={p}_{ji}/{p}_{ii}\end{eqnarray} \tag{ 10 }$$

where K is a constant; k_B is the Boltzmann constant; C_{0 i} and C_{0 j} are temperature independent proportional constants; ε_ij (ε_ij  = ε_ji ), ε_ii and ε_jj are pair potential energy of i-j, i-i and j-j, respectively; p_ij is the probabilities that the molecule i arise in the first coordination layer of the centrical molecule j. A_ij and A_ji are temperature independent in the system of weak intermolecular interaction such as alloys separation during vacuum distillation. However, B_ij and B_ji are temperature dependent and the B_ij (T₁) and B_ji (T₁) at other temperature T₁ are obtained from equation (11).

$$\begin{eqnarray}&&{B}_{ij}\left({T}_{1}\right)=\exp \left(\displaystyle \frac{T}{{T}_{1}}\,\mathrm{ln}\,{B}_{ij}\right)\,\,{B}_{ji}\left({T}_{1}\right)=\exp \left(\displaystyle \frac{T}{{T}_{1}}\,\mathrm{ln}\,{B}_{ji}\right)\end{eqnarray} \tag{ 11 }$$

According to the relationship (∂G_{E m} /∂x_i )_{T, P, xi≠j} = RTlnγ_i , the activity coefficients of i and j are:

$$\begin{eqnarray}&&\mathrm{ln}\,{\gamma }_{i}=-\,\mathrm{ln}\,{D}_{ji}-{x}_{i}{x}_{j}\displaystyle \frac{{C}_{ji}}{{D}_{ji}}+{x}_{j}^{2}\left(\displaystyle \frac{{A}_{ij}+{C}_{ij}}{{D}_{ij}}+\displaystyle \frac{{A}_{ji}}{{D}_{ji}}\right)-{x}_{i}{x}_{j}^{2}\left(\displaystyle \frac{{A}_{ji}\,\cdot \,{C}_{ji}}{{D}_{ji}^{2}}-\displaystyle \frac{{A}_{ij}\,\cdot \,{C}_{ij}}{{D}_{ij}^{2}}\right)\end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray}&&\mathrm{ln}\,{\gamma }_{j}=-\,\mathrm{ln}\,{D}_{ij}-{x}_{i}{x}_{j}\displaystyle \frac{{C}_{ij}}{{D}_{ij}}+{x}_{i}^{2}\left(\displaystyle \frac{{A}_{ji}+{C}_{ji}}{{D}_{ji}}+\displaystyle \frac{{A}_{ij}}{{D}_{ij}}\right)-{x}_{j}{x}_{i}^{2}\left(\displaystyle \frac{{A}_{ij}\cdot {C}_{ij}}{{D}_{ij}^{2}}-\displaystyle \frac{{A}_{ji}\cdot {C}_{ji}}{{D}_{ji}^{2}}\right)\end{eqnarray} \tag{ 13 }$$

where the parameters C_ij , C_ji and D_ij , D_ji are defended as equations (14) and (15).

$$\begin{eqnarray}&&{C}_{ij}=1-\displaystyle \frac{{V}_{mi}}{{V}_{mj}}{B}_{ij}\,\,{C}_{ji}=1-\displaystyle \frac{{V}_{mj}}{{V}_{mi}}{B}_{ji}\end{eqnarray} \tag{ 14 }$$

$$\begin{eqnarray}&&{D}_{ij}={x}_{j}+{x}_{i}\displaystyle \frac{{V}_{mi}}{{V}_{mj}}{B}_{ij}\,\,{D}_{ji}={x}_{i}+{x}_{j}\displaystyle \frac{{V}_{mj}}{{V}_{mi}}{B}_{ji}\end{eqnarray} \tag{ 15 }$$

Separation coefficient β is used for the judgement of separation feasibility of alloys in vacuum distillation [10]. For i-j binary system,

$$\begin{eqnarray}&&{\beta }_{i}=\displaystyle \frac{{\gamma }_{i}}{{\gamma }_{j}}\cdot \displaystyle \frac{{P}_{i}^{* }}{{P}_{j}^{* }}\end{eqnarray} \tag{ 16 }$$

where γ_i and γ_j are the activity coefficients of i and j, respectively; P_*i is the saturated vapor pressure of component i which can be obtained from Van Laar equation [10]:

$$\begin{eqnarray}&&\mathrm{lg}\,{P}_{i}^{* }=A{T}^{-1}+B\,\mathrm{lg}\,T+CT+D\end{eqnarray} \tag{ 17 }$$

where A, B, C and D are evaporation constants; T is the absolute temperature in Kelvin.

The thermodynamic condition of VLE of i-j binary system is the fugacity of component i in liquid phase ${\bar{f}}_{i}^{L}$ equal to that of i in vapor phase ${\bar{f}}_{i}^{V}:$

$$\begin{eqnarray}&&{\bar{f}}_{i}^{L}\left({x}_{i},T,P\right)={\bar{f}}_{i}^{V}\left({y}_{i},T,P\right)\end{eqnarray} \tag{ 18 }$$

If pressure effect can be neglected, in other words, the vapor phase is an ideal gas, ${\bar{f}}_{i}^{V}$ can be expressed as equation (19). In this case, the liquid phase cannot be considered ideal. The deviations from ideality are investigated by introducing a correction factor γ_i into Raoult's law and ${\bar{f}}_{i}^{L}$ can be expressed as equation (20).

$$\begin{eqnarray}&&{\bar{f}}_{i}^{V}\left({y}_{i},T,P\right)={f}_{i}^{V}\cdot {y}_{i}\cdot P\end{eqnarray} \tag{ 19 }$$

$$\begin{eqnarray}&&{\bar{f}}_{i}^{L}\left({x}_{i},T,P\right)={f}_{i}^{* }\,\cdot \,{\gamma }_{i}\,\cdot \,{x}_{i}\,\cdot \,{P}_{i}^{* }\,\cdot \,\exp \left[\displaystyle \frac{{{V}_{i}}^{l}\left(P-{{P}_{i}}^{* }\right)}{RT}\right]\end{eqnarray} \tag{ 20 }$$

where ${f}_{i}^{V}$ is vapor phase fugacity coefficient of i; ${f}_{i}^{* }$ is saturated fugacity coefficient of pure i in liquid phase; V_i ^l is liquid molar volume of component i.

The vapor phase is regarded as an ideal gas in vacuum distillation, thus ${f}_{i}^{V}$ and ${f}_{i}^{* }$ are equal to one and $\exp \left[{{V}_{i}}^{l}(P-{{P}_{i}}^{* })/RT\right]$ equals approximately one at low pressure. Simultaneous equations (18)–(20), the equation of modified Raoult's law can be written as:

$$\begin{eqnarray}&&{y}_{i}\cdot P={\gamma }_{i}\cdot {x}_{i}\cdot {P}_{i}^{* }\end{eqnarray} \tag{ 21 }$$

For i-j binary alloy system:

$$\begin{eqnarray}&&{x}_{i}+{x}_{j}=1{y}_{i}+{y}_{j}=1\end{eqnarray} \tag{ 22 }$$

$$\begin{eqnarray}&&P={P}_{i}^{* }{\gamma }_{i}{x}_{i}+{P}_{j}^{* }{\gamma }_{j}{x}_{j}\end{eqnarray} \tag{ 23 }$$

Combining equations (21)–(23), x_i and y_i are:

$$\begin{eqnarray}&&{x}_{i}=\displaystyle \frac{P-{P}_{j}^{* }{\gamma }_{j}}{{P}_{i}^{* }{\gamma }_{i}-{P}_{j}^{* }{\gamma }_{j}}{y}_{i}=\displaystyle \frac{{P}_{i}^{* }{\gamma }_{i}{x}_{i}}{P}\end{eqnarray} \tag{ 24 }$$

Substituting the corresponding parameters of pure metals (table 1) [22, 23] into equation (6), the coordination numbers Z_i and Z_j of MIVM were obtained. Substituting Z_i , Z_j , the experimental data of activity coefficients (γ_{i, exp.} and γ_{j, exp.}) [24, 25] and the relevant parameters into equations (3) and (4), the pair-potential energy interaction parameters (B_ij and B_ji ) of MIVM were obtained by fitting at least squares principle.

Similarly, the volume parameters (B_ij and B_ji ) and the energy parameters (A_ij and A_ji ) of M-MIVM can be obtained from equations (12) to (15). The calculated model parameters of MIVM and M-MIVM are shown in table 2.

Substituting B_ij , B_ji , Z_i , Z_j and corresponding parameters of MIVM into equation (3) and equation (4), the activities and activity coefficients of components of Ag-Pb, Ag-Sb, Ag-Bi, Au-Pb, Pd-Pb, Pt-Pb and Cu-Pb alloys can be predicted. Similarly, substituting A_ij , A_ji B_ij , B_ji and relevant parameters of M-MIVM into equations (12) to (15), the activities of above binary precious alloys can be calculated.

For the purpose of confirming the dependability of M-MIVM, two other successful classical thermodynamic model Wilson equation and NRTL model were employed to calculate the activities of these binary precious metal alloys. Wilson equation is a two-parameter model and the adjustable parameters A_ij and A_ji can be obtained by utilizing the Newton-Raphson methodology on the condition that the experimental values of activities are known [26]. NRTL model is deduced from the two-liquid solution theory of Scott [27] and local composition concepts of Wilson. It is also regarded as a reliable model to predict the activities of liquid alloys. NRTL model is a three-parameter model and the non-randomness factors α_ji (α_ij = α_ji ) and the energy parameters τ_ij and τ_ij (τ_ij  ≠ τ_ij ) are calculated by the Newton-Raphson methodology under the condition of the experimental values of activity coefficients are known. All of the obtained model parameters of Wilson equation and NRTL model were listed in table 2.

The activities and activity coefficients of these precious metal alloys calculated from MIVM, M-MIVM, Wilson equation and NRTL model are shown in figure 1(a) to (g).

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Figures 1(a) and (b) show that all models display great data fitting capability in Au-Pb and Pd-Pb alloys. The reason is that MIVM, Wilson equation and NRTL model are good at handling symmetric systems (The characteristic of the symmetric system is that the experimental activity curves of the two components intersect near x = 0.5 in the coordinate axis, and are symmetrical about x = 0.5 or near.). M-MIVM inherits this good capacity from MIVM and plays it better. For Ag–Pb, Ag–Bi, Ag–Sb, Pt–Pb and Cu-Pb alloys, figures 1(c) to (g) show that the predicted values of MIVM, Wilson equation and NRTL model are obviously disagreement with the experimental data. However, M-MIVM shows great data fitting capability in these asymmetric systems (The characteristic of the asymmetric system is that the two experimental activity curves are generally not mirror-symmetrical, and the abscissa of the focus of the measurement curve has different degrees of deviation relative to x = 0.5.), especially for the alloy system with complex trend of activity coefficients such as Ag–Pb, Ag–Bi and Ag–Sb alloys. In fact, the complicated change trend is the characteristics of the alloy system and these characteristics will directly affect the VLE calculation.

In order to accurately characterize the deviation extent between experimental values and calculated data and the reliability of M-MIVM, the average standard deviation S_*i and the average relative deviations S_i of each model are also calculated from equation (29), as shown in figure 2 and table 3.

$$\begin{eqnarray}&&{{S}_{i}}^{\ast }=\pm {\left[\displaystyle \frac{1}{n}\displaystyle \sum _{i=1}^{n}{\left({a}_{i,{\rm{pre}}.}-{a}_{i,\exp {\rm{.}}}\right)}^{2}\right]}^{1/2}\,\,{S}_{i}=\pm \displaystyle \frac{100}{n}\displaystyle \sum _{i=1}^{n}\left|\displaystyle \frac{{a}_{i,{\rm{pre}}.}-{a}_{i,\exp {\rm{.}}}}{{a}_{i,exp.}}\right|\end{eqnarray} \tag{ 29 }$$

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Figure 2 shows that the values of S_*i and S_i of M-MIVM were smallest among the four models, indicating that the M-MIVM has higher reliability and universality.

Table 3 shows that in symmetric system (Au-Pb and Pd-Pb), the average relative deviations of each model are low enough, less than 3% in Au-Pb alloy and less than 10% in Pd-Pb alloy, respectively. The differences of the average standard deviations of each model are not very significant. However, the values of S_{* i} and S_i of M-MIVM are lower than that of other three models. In asymmetric system (Ag-Pb, Ag-Bi, Ag-Sb, Pt-Pb and Cu-Pb alloys), the values of S_{* i} and S_i of MIVM, Wilson equation and NRTL model are several times larger than that of M-MIVM. Particularly, for Ag-Sb alloy, the values of S_i of Wilson equation and NRTL model are over 10% and MIVM over 20%, while M-MIVM less than 4%; for Pt-Pb alloy, it is hard to estimate whether it is symmetric or asymmetric system because of the absence of partial experimental data, the values of S_i of MIVM and NRTL model are around 14% and Wilson equation are close to 20%, while M-MIVM is 6.02% and the S_{* i} values of M-MIVM are decreased about three times than that of other three models. The Cu-Pb alloy usually produced with the precious alloys during the recovery process of electronic waste and it is necessary to be studied. It is asymmetric system, and the values of S_{* i} and S_i of M-MIVM are 0.0058 and 0.62%, respectively, which are quite small than that of other three models.

The average relative deviations of M-MIVM in all binary precious metal alloys are less than 7%, which indicates that M-MIVM is reliable and stable. The prediction capacity and effect of M-MIVM are better than other models in symmetric or asymmetric alloy system. M-MIVM with excellent data fitting capability can describe the characteristics of these binary precious metal alloys more accurately which will in turn improve the calculation accuracy of the separation coefficients and VLE data in vacuum distillation. In addition, this excellent capability undoubtedly makes up for the drawback of other classic models in asymmetric systems and will greatly expend the application range of M-MIVM.

The separation coefficient plays a relatively important role on researching these binary precious metal alloys in vacuum distillation. The yields of silver, gold, palladium and platinum are far higher than that of other PMs, and the Ag-Pb, Au-Pb, Pd-Pb and Pt-Pb are the commonest alloys in PMs production and recycling. Therefore, the separation coefficients of Ag-Pb, Au-Pb, Pd-Pb and Pt-Pb alloys were calculated from equation (16) based on M-MIVM, as shown in figures 3(a) to (d). The saturated vapor pressure P_{* i} of each component was calculated form equation (17) and the relevant parameters for P_{* i} calculation of pure metals are presented in table 4.

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The separation coefficients of Ag-Pb, Au-Pb, Pd-Pb and Pt-Pb alloys are much larger than one, indicating that they can be easily separated by vacuum distillation. In addition, the results show that the separation coefficients are decreased with the temperature increasing, indicating that the tendency of liquid phase volatilization is more obvious with the temperature increasing, and that it is not favorable for separation when the temperature too high.

Separation coefficient can judge the feasibility of separation of precious metal alloys by vacuum distillation, but cannot predict the separation degree. Therefore, the VLE phase diagram is needed to be calculated to quantitatively and accurately analysis of the distribution of alloy components in liquid and vapor phase in vacuum distillation.

Substituting corresponding parameters γ_i , γ_j , P_{* i} , and S_{* i} under different temperatures into equations (22) to (24), the VLE data can be obtained, and then T-x-y and P-x-y phase diagram can be drawn. The saturated vapor pressure P_{* i} is calculated from equation (17) and the needed constants are shown in table 4. The flowchart of T-x-y and P-x-y phase diagram calculation is displayed in figures 4(a) and (b), respectively. T-x-y phase diagram calculation is an iterative procedure in which a reasonable and estimated temperature T is needed first, and then the partial pressure P_i can be calculated from ${P}_{i}={P}_{i}^{* }{\gamma }_{i}{x}_{i}$ at the temperature T. When $\left|\left({P}_{i}+{P}_{j}\right)-P\right|/P\leqslant 0.01 \% ,$ the corresponding y_i and T can be obtained. The calculation process of P-x-y phase diagram resembles that of T-x-y phase diagram.

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Following the procedure shown in figure 4, the VLE phase diagrams of Ag-Pb, Au-Pb, Pd-Pb and Pt-Pb alloys were calculated based on M-MIVM, as shown in figures 5 to 8.

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It is founded that the calculation values are basically consistent with the experimental data in figures 5(a), 6(a) and 8(a). These deviations are inevitable and reasonable because of the complex experimental conditions, high temperature and low pressure in the procedure of intermittent and small-scale vacuum distillation.

Besides, the T-x-y phase diagrams show that the temperature fluctuation extent of vapor and liquid phase lines decrease with pressure of system reduction, indicating that the alloys can be separated more easily at lower pressure with less energy consumption. For example, for Au-Pb alloy, if Au purity is required to be larger than 0.999 (mole fraction), when the system pressure is decreased from 15 Pa to 5 Pa, the minimum design temperature can be decreased from 1907.6K to 1771.2 K.

The optimal experimental conditions of separation can also be acquired through T-x-y phase diagrams. For instance, if the precious metal purity is required to be larger than 0.999 (mole fraction), the temperature of separation is needed to be higher than 1369.7 K for Ag-Pb alloy, 1771.2 K for Au-Pb, 2176.9 K for Pd-Pb and 2304.7 K for Pt-Pb at 5 Pa.

Furthermore, the law of leverage can also be utilized into VLE phase diagram analysis to quantificationally forecast the amount of substance of distillate and residue under the condition of the separated temperature and pressure are determined. Take T-x-y phase diagram of Ag-Pb alloy as an example, suppose x_E is the mole fraction of Pb in Ag-Pb binary system under the condition of the separation pressure and temperature of the system are 5 Pa and 1200 K, respectively. And then make a straight line that intersects the liquidus and gaseous lines at points A and B, respectively, as shown in figure 5(a). The components of A and B are x_l and y_g , respectively when the alloy system reaches VLE. According to the law of leverage:

$$\begin{eqnarray}&&\displaystyle \frac{{n}_{l}}{{n}_{g}}=\displaystyle \frac{{y}_{g}-{x}_{E}}{{x}_{E}-{x}_{l}}=\displaystyle \frac{\left|EB\right|}{\left|EA\right|}\end{eqnarray} \tag{ 30 }$$

where n_l and n_g are the amount of substance of residues and volatiles, respectively. $\left|EA\right|$ and $\left|EB\right|$ are the length of line of segment EA and EB, respectively.

If the total moles of raw materials are n (n = n_l + n_g ), then the moles of residues and volatiles (n_l and n_g ) can be calculated from equations (31) and (32), respectively:

$$\begin{eqnarray}&&{n}_{l}=\displaystyle \frac{{y}_{g}-{x}_{E}}{{y}_{g}-{x}_{l}}n=\displaystyle \frac{\left|EB\right|}{\left|AB\right|}n\end{eqnarray} \tag{ 31 }$$

$$\begin{eqnarray}&&{n}_{g}=\displaystyle \frac{{x}_{E}-{x}_{l}}{{y}_{g}-{x}_{l}}n=\displaystyle \frac{\left|EA\right|}{\left|AB\right|}n\end{eqnarray} \tag{ 32 }$$

where $\left|AB\right|$ is the length of AB.

The P-x-y phase diagrams of Ag-Pb, Au-Pb, Pd-Pb and Pt-Pb alloys with a certain temperature range were obtained based on M-MIVM. The P-x-y phase diagrams can be used to analyze the product component dependence of temperature or pressure in the process of vacuum distillation. In other words, if the temperature and system pressure of operation are selected based on VLE phase diagrams, the metal content in residue and volatile can be predicted.