THERMODYNAMIC ANALYSIS OF PHASE DIAGRAMS BASED ON THE CONCEPT OF THE BJERRUM–GUGGENHEIM OSMOTIC COEFFICIENT

The aim of this study is to theoretically confirm or refute the high degree of lead and zinc volatilization from the Shalkiya deposit (polymetallic refractory ores) using analyzing the phase diagrams and the behavior of the osmotic coefficient according to the Bjer-rum-Guggenheim equation along the liquidus line for binary systems based on calcium. From the plots depicting the dependence of the Bjerrum-Guggenheim osmotic coefficient on activity, one can infer the structure of the melt (positive Ф_i <1 or negative Ф_i >1). That is, if the values of the Bjerrum-Guggenheim osmotic coefficient in a binary system indicate a positive character of interaction between the components, it implies interaction between like atoms in the melt, whereas negative values indicate interaction between unlike atoms, suggesting the formation of complex, high-temperature compounds, which would be undesirable in our case. The formation of complex high-temperature chemical compounds with impurity elements such as zinc and lead in the Fe-Si-Al-Ca-Mg-Zn-Pb system remains unresolved.


INTRODUCTION
In recent years, there has been an increasing number of publications addressing the issue of raw materials in metallurgy.The reserves of high-grade and easily processable raw materials worldwide are steadily decreasing.The forefront of research is occupied by theoretical and applied studies in the field of processing refractory polymetallic ores, particularly those from the Shalkiya deposit.Most multi-component lead-zinc and copper-zinc ores belong to the class of refractory ores [1] - [8].The processing of refractory polymetallic ores is characterized by its complexity due to the chemical as well as physical composition of those ores.In addition, difficulties arise due to the lack of real understanding of the phase composition of such raw materials.

MATERIAL AND METHODS
To gain insights into the behavior of zinc and lead, particularly whether complex high-temperature compounds will form in complex alloys, the main crystallization regions of binary systems based on iron, silicon, aluminum, calcium, and magnesium are mathematically described using unified analytical frameworks.The variation in the Bjerrum-Guggenheim osmotic coefficient as a function of the components' activity ratio in the ideal liquid and solid phases (positive Ф_i <1 or negative Ф_i >1) under boundary conditions governing the formation of crystallization fields of phases related to the processes of complex alloy smelting is direct evidence that zinc and lead in the melt will form volatile fractions.In order to compute numerical values of the Bjerrum-Guggenheim osmotic coefficient and construct dependency plots (Ф_i) as a function of activity, initial data for 5 binary phase diagrams with calcium (Fe-Ca, Ca-Pb, Ca-Zn, Ca-Mg, Ca-Al, Ca-Si) have been formulated, constituting a comprehensive 7-component system of Fe-Si-Al-Ca-Mg-Zn-Pb.All phase diagrams featuring the formation of congruently melting compounds can be divided into two groups based on their shape (radius of curvature) with sharp and low-pitched maxima.Sharp (singular) maxima of liquidus and solidus curves in both liquid and solid phases correspond to perfectly non-dissociating compounds.The radii of curvature of the liquidus lines at these points are zero.If dissociation occurs in both phases, both maxima are smooth.We conducted a theoretical analysis of the behavior of Bjerrum-Guggenheim coefficient plots near the melting temperatures of the corresponding chemical compounds.

RESULTS AND DISCUSSION
The obtained results allow us to conclude that the proposed method is universally applicable when it comes to analytically describing phase diagrams from the perspective of the Bjerrum-Guggenheim concept [9] - [12].The available experimental results on binary phase diagrams contain two coordinates for each point on the liquidus and solidus lines, namely: the concentration of the crystallizing component Х  () and the temperature K.
To generate the initial data and construct the plots depicting the dependence of the Bjerrum-Guggenheim coefficient (Ф  ′ и Ф  ′′ ) on the activity ratio in liquid and solid phases according to the Schroeder-Le Chatelier equation: ] ,  ,,1  (1.) experimental data from various model binary systems were used to obtain these values at a specific temperature   .according to the following equations: and   = ln ( 1  / 1  ) ) where  1  / 1  is the concentration of the crystallizing component at a given temperature;  1  / 1  is the activity of the crystallizing component for an ideal solution, calculated according to the equation (1).The essence of the method lies in determining the Bjerrum-Guggenheim coefficient (Ф  ′ и Ф  ′′ ) along the liquidus and solidus lines in the form of dependencies listed below: ) ) These, in turn, made it possible to derive semi-empirical dependencies for calculating the liquidus and solidus lines of each crystallizing phase of the system under consideration: ) where Нm,1(2) is the melting enthalpy of the 1 st and the 2 nd components at the melting temperature, J/mol; R is the universal gas constant, 8.3144 J/mol K; Tm,1(2) is the melting temperature of the 1 st and the 2 nd components and crystallization of the melts, K; T is the current temperature along the liquidus line, K; Ф  ′ is the Bjerrum-Guggenheim coefficient for the 1st and the 2nd components, which allows one to find the correlation dependence and obtain a mathematical expression for the ratio of activities of the i th component in the liquid and solid phases according to equation (3); Ф  ′′ is the Bjerrum-Guggenheim coefficient for the 1st and the 2nd components, which allows one to find the correlation dependence and obtain a mathematical expression for the liquidus line of the i th component according to equation (4).For the dependencies (3,4), the equation coefficients are computed using the method of least squares, with the establishment of the correlation coefficient Rxy (for linear dependencies) or through the convergence dispersion σ (for nonlinear dependencies).All calculations are performed using programs written in the Delphi language.With the programs developed by us, phase equilibria of numerous binary systems were studied.Processing the obtained results using the method of least squares allows to find the values of constants and correlation coefficients for various systems and deriving analytical expressions for phase crystallization fields for any quasi-system (for the partial system).In addition, based on the real possibility of assessing the behavior of compounds in the melt and the nature of interparticle in-teractions in the liquid phase using the Bjerrum-Guggenheim osmotic coefficient (Ф  ), the analysis of the phase crystallization lines also allows for determining the reliability of data on the phase diagrams of these systems and the stability of the compounds in the melts.This methodology enables thermodynamic assessment of the mutual solubility of components in each other based on the nature of the change in the Bjerrum-Guggenheim osmotic coefficient.

Ca -Pb phase diagram
There are four intermetallic compounds in the system, two of which, CaPb3 and Ca2Pb, melt congruently at 666 and 12050C, respectively.CaPb and Ca5Pb3 compounds are formed by peritectic reactions at 908 and 1127 0 C, respectively (Fig. 1).The reference data from [13] generated initial data on simple and complex components' enthalpies and melting temperatures.The enthalpies of substances for which data could not be found were calculated.Four crystallization areas were considered in this system, i.e. two for the partial systems Pb3Ca-Pb and Pb3Ca-Ca, and two partial systems for the relatively congruent compound Ca2Pb: Ca2Pb-Pb Ca2Pb-Ca.The first Pb3Ca crystallization region in the Pb3Ca-Pb system (Fig. 1) is quite extensive.It starts from the melting point of Pb3Ca (939 K) and goes through to the melting temperature of lead (600 K) [14] - [19].The procedure for mathematical description of monovariant phase equilibrium lines on the phase diagram based on the concept of the Bjerrum-Guggenheim osmotic coefficient is described in [9] - [12].For thermodynamic calculations, the enthalpy of melting was assumed to be equal to ∆H m,CaPb3 == 7450 J/mol at a melting temperature of 939 K [14], [15], [19].

Fig. 1 Ca-Pb phase diagram
Table 1 presents the initial data on the crystallization of Pb3Ca by temperature and its corresponding composition up to the eutectic line at 1588 K. for the base system (Fig. 2) in order to see the behavior of the components in the melt.
We constructed a new plot of the dependence of the Bjerrum-Guggenheim osmotic coefficient for a partial system (Fig. 3).The mathematical expression of the Pb3Ca crystallization line in the Pb3Ca-Pb system is given in the form of a semi-empirical dependence of the Schroeder-Le Chatelier equation (9): , (9.) Table 2 shows the results on the convergence of the calculated (9) and experimental data (  We will now consider the Pb3Ca crystallization region for the partial Pb3Ca-Ca system.We conducted the same manipulations as in the first area.We collected data on the phase diagram (Fig. 1) for temperatures ranging from 939 through to 911 K and the corresponding composition.The initial data is presented in Table 3.For thermodynamic calculations, the melting enthalpy was assumed to be equal to HmCaPb3 = 7450 J/mol [13], [14], [15] at a melting temperature of 939 K [8].
Table 3 Initial data of the Pb3Ca crystallization region for the partial Pb3Ca-Ca system T,K What is characteristic of this particular Pb3Ca crystallization region?The graph exhibits a strictly correlated relationship with a correlation coefficient of -0.998: ) relative to the basic Pb-Ca system.This means that this Pb3Ca crystallization region is characterized by the presence of van der Waals interaction forces between the components.Using the Schroeder-Le Chatelier equation for an ideal system (9) and equation (10), which characterizes the deviation of the properties of a real system from an ideal one, we obtained the calculated values of the   components according to the mathematical relationship (11): ) Then, the obtained results (11) were compared with the experimental values in  The conclusion regarding the congruent compound Pb3Ca crystallization region for the two partial systems, i.e.Pb3Ca-Pb and Pb3Ca-Ca, is as follows: from the lead side of the Pb3Ca-Pb partial system, dissociation of the compound predominantly occurs (the Bjerrum-Guggenheim coefficient exhibits a positive deviation from ideality), and the degree of dissociation increases as the temperature changes further.On the other hand, considering the calcium side of the Pb3Ca crystallization region, van der Waals forces predominantly govern the interactions between the components.
We will now consider the high-temperature congruent compound Ca2Pb crystallization region.For thermodynamic calculations, we adopted the enthalpy of fusion as HmCa2Pb = 12400 J/mol [14] at the melting temperature of 1478 K [19].Similar to obtaining data on the melt structure in the Ca2Pb crystallization region, we collected initial data from the phase diagram (Fig. 1) for the temperature range of 1478-911 K and the corresponding composition.The initial data are presented in Table 5.
A graph (Fig. 5) depicting the dependence of the Bjerrum-Guggenheim coefficient on activity was constructed for the baseline data (without conversion to the formation of Ca2Pb).It is visually apparent that the graph exhibits a nonlinear relationship, indicating either the formation of associates of varying complexity in accordance with the compounds present on the diagram, or dissociation of the compound, or the formation of dimers, trimers, etc. in the crystallization region of pure elements.Additionally, the values of the Bjerrum-Guggenheim coefficient from activity are less than one, indicating interaction between like atoms in the melt, hence an intense dissociation process is occurring.In order to obtain the mathematical dependence of the Bjerrum-Guggenheim osmotic coefficient on activity, the least squares method was employed (12): The mathematical expression of the Ca2Pb crystallization line in the Ca2Pb -Pb system is given in the form of a semi-empirical dependence of the Schroeder-Le Chatelier equation ( 13): (13.)According to ( 12) and ( 13), a comparative analysis of the convergence of the calculated and experimental data was conducted and its results are listed in Table 6.The first noticeable observation when analyzing the convergence of the experimental and calculated compositions is the lack of convergence from the melting temperature of Ca2Pb up to 1241 K. Within this temperature range, the formation of the incongruent Ca5Pb3 and CaPb compounds occurs, indicating their strong influence on the congruent Ca2Pb compound's formation.
We constructed a graph showing the dependence of the Bjerrum-Guggenheim osmotic coefficient Ф оп ′′ on activity for the partial Ca2Pb-Pb system (Fig. 6).The calculated compositions adjusted for the formation of the congruent Ca2Pb compound are provided in Table 5.From the melting temperature of 1478 K to the formation of the first incongruent Ca5Pb3 compound, the graph experiences a sharp surge, and little can be done about it.According to the cal-culated data, the influence of the second incongruent CaPb compound on the graph's position is not significant but still considerable.Given the substantial impact of incongruent compounds, we recalculated the data to account for their formation and observed how the graph of the Bjerrum-Guggenheim osmotic coefficient would change.Table 7 shows the composition of Ca2Pb considering the formation of Ca5Pb3 and CaPb.It's worth noting that Table 7 provides data for the Bjerrum-Guggenheim osmotic coefficient for the baseline system Ф .
′′ .Negative values of the coefficient from the melting temperature up to the eutectic point only increase.The curvature of Φi is primarily influenced by the nature of the directional bonds in the associates and the change in their numerical values with temperature, as well as the degree of dissociation of the compound or its stabilization, even transitioning smoothly to association processes as the temperature changes further.Fig. 7 shows a graph of the dependence of the Bjerrum-Guggenheim osmotic coefficient (Ф оп ′′ ) on the activity of Ca2Pb crystallization of the private Ca2Pb -Pb system, taking into account the formation of incongruent compounds Ca5Pb3 and CaPb.As we can see from the picture, it has become strictly linear.Using the Schroeder-Le Chatelier equation for an ideal system and equation ( 14), which characterizes the deviation of the properties of a real system from an ideal one, we obtained the calculated values of the components X L and X S according to the mathematical relationship (15): ) Then the results obtained using (15) were compared with the experimental values.Experimental and calculated data are given in Table 8. from the table for the partial Ca2Pb-Ca system, we plotted the dependence of the osmotic coefficient on the activity ratio (Fig. 8).The resulting graph in Fig. 8 is clearly convex, which means that an intense dissociation of the congruent Ca2Pb compound occurs in the melt.The calculated osmotic coefficient values for a nonlinear relationship were found using the least squares method in the form of equation ( 16): The mathematical expression of the Ca2Pb crystallization line in the Ca2Pb-Ca system is given in the form of a semi-empirical dependence of the Schroeder-Le Chatelier equation (17).The conclusions regarding the crystallization region of the Ca2Pb-Pb and Ca2Pb-Ca partial systems are as follows: the structure of the melt is not as simple as it seems at the first glance.Graphs of the dependence of the Bjerrum-Guggenheim osmotic coefficient (Ф  2  ′ ) on the Ca2Pb crystallization activity a L Ca2Pb for the partial Ca2Pb-Pb and Ca2Pb-Ca systems even in terms of partial systems clearly demonstrate a nonlinear character.This indicates that besides the high-temperature congruent compound, the melt also contains and is strongly influenced by the presence of incongruent Ca5Pb3 and CaPb compounds [20], [21].Negative deviations of the osmotic coefficient from the ideal state further confirm the possibility of forming associations of varying complexity in the melt.

𝐿
for the partial CaZn2-Ca system relative to the baseline data.The Bjerrum-Guggenheim osmotic coefficient for this CaZn2 crystallization region has negative values.Additionally, the dependence in Fig. 10 is curvilinear, which means that in the melt, the curvature Ф  is primarily influenced by the nature of the directional bonds in the associates and the change in their numerical values with temperature.The graph of the dependence of the Ф  2 ′ on   2  exhibits weak concavity, starting from the melting temperature of CaZn2.This suggests a strong interparticle interaction of the components in the melts and weak dissociation of CaZn2.However, with the transition of compositions to the CaZn2 crystallization region, the formation of associates occurs, as the graph of Ф  2 ′ has a concave character in this area.By processing the data using the least squares method according to the dependence of the Ф  2 ′ on   2

𝐿
, we obtain the dependence (18), with the calculation result =0.0086(which gives a deviation from the experimental data within 0.2% abs.) for the liquid-phase of the CaZn2 crystallization region.
) Fig. 10 The graph of the dependence of the Bjerrum-Guggenheim osmotic coefficient (Ф оп ′ ) on the CaZn2 crystallization activity a L CaZn2,id.*for the baseline Ca-Zn system A critical analysis of the convergence of the calculated and experimental data is presented in Table 12.Table 12 Comparative analysis of the the calculated and experimental data for the region of CaZn2 crystallization relative to the basic Ca-Zn system according to equation ( 18 for the partial CaZn2-Ca system was plotted (Fig. 11).(19) and experimental data (Table 11) is presented in Table 13.CaZn2 crystallization region in the CaZn2-Zn system.Employing the procedure described above, Table 14 shows the initial data for the CaZn2-crystallization region in the CaZn2-Zn system with conversion factors for this region equal to I=1, J=2, Q=0 P=1.
To calculate the activity ratio in the liquid and solid phases according to (3), the following thermodynamic data were used: ∆HCaZn2=9230 J/mol, Tmel.CaZn2= 977 K [14], [15].clearly exhibits a convex nature, starting from the melting temperature of the chemical compound and increasing as the temperature decreases to the eutectic horizontal at 915 K, indicating an increase in the dissociation process of the congruent compound (Fig. 12).Table 15 shows the convergence of the experimental concentrations with the ones calculated using (20).should exhibit a strictly correlational form [27].

CONCLUSION
Theoretical analysis of the melt structure based on the behavior of the Bjerrum-Guggenheim osmotic coefficient in the melts of the general Fe-Si-Al-Ca-Mg-Zn-Pb system showed that complex high-temperature chemical compounds with impurity elements such as zinc and lead do not form in the melt, and there are no obstacles to the high degree of volatility of these elements.Theoretical studies have been conducted on the nature of changes in the Bjerrum-Guggenheim osmotic coefficient (positive Ф  <1 or negative Ф  >1 in the melt) for two binary Ca-Pb and Ca-Zn systems.Phase diagrams of Ca-Pb and Ca-Zn are characterized by the formation of congruent compounds.The behaviour of the Bjerrum-Guggenheim osmotic coefficient for systems containing calcium shows that almost all systems exhibit positive deviation from ideality, i.e., interactions between atoms of the same type occur.Congruent compounds are characterized by a high degree of dissociation.

Fig. 2
Fig. 2 Dependence of the Bjerrum-Guggenheim osmotic coefficient Ф оп ′′ on the activity ratio S Ca Pb L Ca Pb a a 3 3

Fig. 4
Fig. 4 The graph of the dependence of the Bjerrum-Guggenheim osmotic coefficient Ф оп ′′ on the Pb3Ca crystallization activity   3  of the partial Pb3Ca-Ca system

Fig. 5
Fig. 5 Graph of the dependence of the Bjerrum-Guggenheim osmotic coefficient Ф оп ′′ on the Ca2Pb crystallization activity  2  of the baseline Ca -Pb system

Fig. 6
Fig. 6 Graph of the dependence of the Bjerrum-Guggenheim osmotic coefficient Ф оп ′′ on the Ca2Pb crystallization activity  2  of the partial Ca2Pb -Pb system

Fig. 7
Fig. 7 Graph of the dependence of the Bjerrum-Guggenheim osmotic coefficient (Ф оп ′′ ) on the activity а  2   crystallization of Ca2Pb of the private system Ca2Pb -Pb, taking into account the formation of incongruent compounds Ca5Pb3 and CaPb

Fig. 8
Fig. 8 Graph of the dependence of the Bjerrum-Guggenheim osmotic coefficient (Ф оп ′′ ) on the Ca2Pb crystallization activity   2   for the partial Ca2Pb -Ca system

Fig. 9
Fig. 9 Ca-Zn phase diagramFor this phase diagram, all crystallization regions of congruent compounds were not described by mathematical expressions because all seven compounds in the melt dissociate before or after the melting temperature.The congruent CaZn2 compound is of greater interest.The thermodynamic data used in the calculations and the derivation of the osmotic coefficient dependency (Ф оп ′′ ) on the for CaZn2 activity   2 ,

Table 15
Comparative analysis of the calculated and experimental data on the CaZn2 crystallization region for the partial CaZn2-Zn system according to(20) both partial systems, i.e.CaZn2-Ca and CaZn2-Zn, the CaZn2 crystallization regions exhibit a distinctly convex character in the graphs depicting the dependence of the Bjerrum-Guggenheim osmotic coefficient (Ф оп ′′ ) on the activity   2 , *  .This indicates that within the range based on the chemical compound CaZn2, solid solutions contain an excess of one of the components relative to the stoichiometric composition of the compound.Obviously, all solid solutions in the region of the intermediate phase homogeneity are unsaturated.Since the left part of the CaZn2-Ca phase diagram occupies a larger area below the ordinate of the congruent compound, and therefore the substitution process is more intense, the graphs depicting the dependence of the Bjerrum-Guggenheim coefficient immediately showed that the dissociation process is ongoing, and the interaction forces between the components are changing.Taking into of dissociation of this compound, the graphs Ф  2 ′′

Table 1
Initial data for the Pb3Ca crystallization region in the Pb-Ca system

Table 2
Comparative analysis of the calculated and experimental data for the Pb3Ca crystallization region in the Pb3Ca-Pb system

Table 3 .
The experimental and calculated data are given in Table4.

Table 4
Comparative analysis of the calculated and experimental data for the Pb3Ca crystallization region in the Pb3Ca-Ca system

Table 5
Initial data on the Ca2Pb crystallization region of the Ca -Pb system

Table 6
Comparative analysis of the calculated and experimental data for the Ca2Pb crystallization region in the Ca2Pb -Pb system

Table 7
Initial data for the Ca2Pb crystallization region for the Ca2Pb -Pb system, taking into account the formation of incongruent compounds Ca5Pb3 and CaPb

Table 8
Comparative analysis of the calculated and experimental data for the Ca2Pb crystallization region of the partial Ca2Pb-Pb system, taking into account the formation of incongruent Ca5Pb3 and CaPb compounds Next, we processed the Ca2Pb crystallization area for the partial Ca2Pb-Ca system.We also took the initial data on the melting point and the corresponding composition.The initial data is shown in

Table 9
Initial data on the Ca2Pb crystallization region for the partial Ca2Pb-Ca system

Table 9 )
were used to analyze the convergence of calculated and experimental data.The results are presented in

Table 10
Comparative analysis of the calculated and experimental data for the Ca2Pb crystallization region in the partial Ca2Pb-Ca system

Table 11
Initial data of the CaZn2 crystallization region for the partial CaZn2-Ca system

Table 13
(19)arative analysis of the calculated and experimental data on the CaZn2 crystallization region for the partial CaZn2-Ca system according to(19)