A theoretically justified method of mathematical description of phase equilibrium lines based on the regularity of the behavior of components along the phase crystallization line has been developed. When analyzing the state diagrams of various systems, a regularity was found in the formation of crystallization fields of phases in the form of a correlation dependence between the osmotic coefficient of the crystallizing component and the ratio of its activity in the liquid and solid phases. The Bjerrum-Guggenheim osmotic coefficient can serve as a measure of the deviation of the energy properties of a real system from the ideal one described by the Schroeder-Le Chatelier equation. Two types of generalized mathematical expression in the form of a modified Schroeder-Le Chatelier equation are proposed for the analytical description of the liquidus and solidus lines of state diagrams of a whole class of systems. These types make it possible to calculate the dependence between the composition and the temperature. Mathematical expressions describing the lines of monovariant phase equilibria are presented on a single analytical basis. Mathematical expressions represent the liquidus and solidus lines of the Fe-Al system for the corresponding elements and compounds using the Schroeder-Le Chatelier equation and the Bjerrum-Guggenheim coefficient (Фi). The linear nature of the change in the osmotic coefficient of crystallizing phases (Фi) on the ratio of the activity of the components in the ideal liquid and solid phases under the boundary conditions of the formation of crystallization fields of the phases was revealed. The type of change in the Bjerrum-Guggenheim osmotic coefficient depends on the intermolecular interaction of the components in the melt. If only van der Waals forces of interaction between the structural units of the considered component with the second prevail, then a linear relationship is observed. When the formation of various groups from the initial components or the dissociation of a chemical compound takes place in the melt, the osmotic coefficient has a curved dependence. This research is funded by the Science Committee of the Ministry of Education and Science of the Republic of Kazakhstan (Grant No. AP08855453).

This article describes a new quantitative evaluation method for the degree of dissociation of a compound depending on the curvature of the liquidus line curve. The main point of this method was to determine the specific expressions for dissociation parameters according to the mathematical model of the crystallization line of the compound on the phase diagram. This was based on generally accepted thermodynamic relations. Empirically, it has now been established that for many types of phase equilibria, a common feature is the presence of a correlation dependence between the ratios of the real Gibbs energy of the distribution of components to its ideal component. The osmotic coefficient of Bjerrum–Guggenheim was used as a measure of the deviation of the system from ideality. This coefficient can be in an analytical form depending on the temperature and composition of the phases. The obtained correlation dependence of the osmotic coefficient of Bjerrum–Guggenheim on the ratio of the activity of the liquid and solid phase has been used to develop a mathematical apparatus to analytically describe the lines and surfaces of the crystallization phases. Accordingly, a single analytical basis, as a universal dependence of the modified Schröder–Le Chatelier equation, has been applied. Based on this equation, the conditions have been defined to develop an equilibrium method with which to calculate the thermal dissociation of chemical compounds. The principle of this method has been examined based on the two-component iron-titanium system. The numerical results of the degree of dissociation of the congruent Fe₂Ti compound have been obtained using the Gibbs energy of the dissociation reaction and the reaction constant. However, it has been found that the degree of dissociation in the compound Fe₂Ti was 0.04%. Demonstrative material for the behavior of the osmotic coefficient of Bjerrum–Guggenheim under boundary conditions has been presented as an assessment criterion of melt structures. The diagrams of the Φi function of the crystallizing phases near the melting temperature of congruently melting compounds (Tm.) have been mathematically studied. Thus, this study demonstrates that Φ${}_{A_m{B}_n}^{\prime }$ diagrams tended to zero and approached the melting temperature of the compound ${\mathrm{T}}_{m,A_m{B}_n}^{}$, i.e., above the melting temperature of this compound Φ${}_{A_m{B}_n}^{\prime }$→∞; then,$\ln x_{A_m{B}_n}$→0, and, as a result, $x_{A_m{B}_n}$→1.

At present time, taking into account the existing situation in the raw material base of production of manganese ferroalloys, the technologies for obtaining new carbon reducing agents for metals from weakly coking, long-flaming and high-ash coals instead of metallurgical coke are developed intensively. It was caused first of all by the fact, that chemical compositions vary cardinally and become low-melting during the process of development of deposits (in particular manganese ores). As a result, technological violations and decrease of production volumes of manganese alloys occur in the furnaces of manganese production shops. Respectively, establishment of regularities in phases interaction, in forming of their crystallization fields and in reducing reactions of the elements by carbon from liquid melts in multi-component systems is especially important. At the same time, the obtained results allow to optimize their manufacturing technologies from different kinds of raw materials and to predict searching ways or rational charge compositions for processing of different low-melting manganese and other raw materials. The aim of this research is to reveal the features of forming of the crystallization fields of galaxite phase MnO–Al2O3 as the most important metallurgical phase and to determine dissociation degree in molten state, as well as to assess influence of these appearances of completeness of extraction of the aimed elements (Mn) from raw materials into metal to stabilize the technological process. The numerical results of dissociation degree for galaxite congruent compound were obtained using Gibbs energy of dissociation reaction and equilibrium constant of this reaction. Material on behaviour of the osmotic coefficient of Bjerrum-Guggenheim is demonstrated as a criterion of melt structure assessment. It was established that dissociation degree of MnO–Al2O3 compound increases with temperature rise and makes 50 % at the melting temperature.This study was made under the project of Committee of Science of the Ministry of Education and Science of the Republic of Kazakhstan for 2021–2023, IRN AP09259157/SPh.