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.

When analyzing a number of phase equilibrium diagrams of various types of systems, the regular relationship was found during the formation of phase crystallization fields in the form of a correlation dependence of the osmotic coefficient of a crystallizing component on the ratio of its activity in the liquid and solid phases. The Bjerrum-Guggenheim osmotic coefficient serves 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. Mathematical expressions for the liquidus and solidus lines are obtained in the form of semi-empirical dependences on a single analytical basis, which make it possible to calculate the temperature dependence of the composition. Thus, a theoretically substantiated method for the mathematical description of the lines of monovariant phase equilibria has been developed, based on the regularities of the behavior of components in the melt. Mathematical expressions represent the crystallization fields of the phases of the MnO-SiO2 system for the corresponding components and compounds using the Schroeder-Le Chatelier equation and the correlation dependence of the osmotic coefficient Ф', experimental and calculated data of the Bjerrum-Guggenheim osmotic coefficient (Фi) are calculated. The type of variation 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 components in the melt prevail, then a correlation dependence is observed. When groups from the initial compounds are formed in the melt, or processes of dissociation or, conversely, association occur, the osmotic coefficient is described by a curvilinear dependence. The paper presents mathematical expressions for the fields of crystallization of phases of the MnO-SiO2 system in the form of semi-empirical dependencies. The nature of the change in the Bjerrum-Guggenheim osmotic coefficient of the crystallizing component depending on the ratio of its activity in the liquid and solid phases is shown. This research is funded by the Science Committee of the Ministry of Education and Science of the Republic of Kazakhstan (Grant No. AP09259157).

A priority direction of ferrous metallurgy development is to increase in output of the high quality metal and metal products of new assortment. One of the methods to improve a quality of steels is to involve of complex alloys based on aluminum, silicon, manganese, etc. for their output. They are necessary as deoxidizing agents and alloying additives. This paper considers the possibility of the thermodynamic substantiation of the aluminum solubility in the ferrosilicon-aluminum complex alloy (FeSiAl) on the basis of their phase diagrams using the osmotic coefficient of the Bjerrum-Guggenheim. Methodology used is based on the theoretical studies of the phase equilibria using the Bjerrum-Guggenheim concept. It includes a set of computer programs in C# language (C sharp) designed to evaluate a deviation scope of properties of a real system from the ideal one. Criterion for evaluation is an osmotic coefficient of the Bjerrum-Guggenheim. The pattern of change in an osmotic coefficient of the Bjerrum-Guggenheim on the ratio of activity of components in the ideal liquid and solid phases (positive Фi 1) under the boundary forming conditions of crystallization regions of phases related to the melting ferrosilicon-aluminum processes is a direct proof of the possibility to use it as a metal reducing agent. The calculated mathematical dependences of the osmotic coefficient of the Bjerrum-Guggenheim permit us to determine the crystallization temperature of the ferrosilicon-aluminum alloy. The alloying process with rich aluminum content is observed at this temperature. The dependence diagrams of an osmotic coefficient of the Bjerrum-Guggenheim of a crystallizing component on the ratio of its activity in the liquid and solid phases demonstrated that a temperature rise leads to strong negative deviations in silicon properties, and thus to its good mixability in the melt with iron and aluminum. Compositions of silicon-aluminum alloys with high aluminum content in the ferrosilicon-aluminum complex alloy (FeSiAl) were determined on the basis of their phase diagrams using the osmotic coefficient of the Bjerrum-Guggenheim with iron content of 12-37%, aluminum 20-25% and silicon 68-38%. The received theoretical results permit to determine conditions which give the maximum possible aluminum assimilation with the ferrosilicon-aluminum melts supplied from the high-ash coal in the melting process of this metal in the ore-thermal furnaces. Thus it is a direct method to develop the output technology of the complex alloys.

This study presents a thermodynamic analysis of the dissociation and association behavior of the Fe–Si system using the Bjerrum–Guggenheim osmotic coefficient. An equilibrium thermodynamic approach was applied to evaluate the Gibbs free energy, equilibrium constant, and degree of association of the congruently melting compound FeSi over a wide temperature range. The Fe–Si system was analyzed across three characteristic crystallization regions: Fe-rich, FeSi, and Si-rich. It was established that the Fe-rich region exhibits behavior approaching ideality with a nearly linear dependence of the osmotic coefficient, whereas the Si-rich region is characterized by strong deviations from ideality due to intensive association processes. The FeSi crystallization region represents a transitional regime in which association and dissociation processes occur simultaneously. The formation and partial dissociation of [Fe_xSi_y] clusters significantly affect the thermodynamic behavior of the melt. It was shown that accounting for FeSi dissociation leads to a linearization of the osmotic coefficient dependence and improves the accuracy of thermodynamic description. The proposed analytical approximations demonstrate high correlation coefficients (R² ≈ 0.99), confirming the reliability of the developed approach. The results provide a consistent thermodynamic framework for describing phase transformations and structural evolution in Fe–Si melts and can be applied to the optimization of metallurgical processes involving silicon-containing alloys.

Недостаточная обоснованность анализа фазовых равновесий является причиной слабого использования обширной термодинамической информации, заключенной в диаграмме состояния. На примере системы железо-кремний показана применимость разработанной методики расчета линий фазовых равновесий для оценки структуры расплава через степень диссоциации или ассоциации конгруэнтно плавящихся химических соединений. Показано, что если в расплаве присутствует явление диссоциации конгруэнтно плавящегося соединения, то полученные новые данные для осмотического коэффициента Бьеррума-Гуггенгейма с учетом степень диссоциации или ассоциации этого соединения в расплаве приобретают прямолинейный характер зависимости. Характерной особенностью линии кристаллизации соединения FeSi является наличие образования ассоциатов (первоначальные графики Фi имели вогнутый характер изменения). Цель работы - выявить закономерные особенности поведения кремния вдоль линии кристаллизации с позиции осмотического коэффициента Бьеррума-Гуггенгейма и получить универсальное математическое выражение для прогнозирования фазового состава сплава при различных концентрациях кремния через степень диссоциации (ассоциации) конгруэнтного соединения FeSi. Впервые предложено математическое выражение линии кристаллизации кремния в системе железо–кремний в виде полуэмпирической зависимости уравнения Шредера-Ле-Шателье, связывающей температуру ликвидуса с концентрацией кремния. На примере системы железо-кремний проведен обоснованный выбор функционального вида аналитического описания поверхности кристаллизации кремния с применением коэффициента Бьеррума - Гуггенгейма. Осуществлена аппроксимация экспериментальных данных диаграммы состояния Fe–Si с оценкой точности и области применимости полученной модели. Выявлены характерные особенности поведения линии кристаллизации кремния, связанные с изменением механизмов кристаллизации и взаимодействием компонентов. Полученные результаты в перспективе могут быть в численном моделировании кристаллизации и прогнозировании свойств железокремнистых сплавов. Ключевые слова: диаграмма состояния, степень ассоциации (диссоциации), линии фазовых равновесий, энергия Гиббса, коэффициент Бьеррума-Гуггенгейма, поля кристаллизации фаз.

The aim of this work is to validate a mathematical approach for describing the liquidus and solidus lines in the Mg–Si system based on the behavior of the Bjerrum–Guggenheim osmotic coefficient in the melt, in order to reveal the nature of interatomic interactions in liquid magnesium alloys. It is shown that, for the base Mg–Si system in the crystallization region of the congruently melting compound Mg2 Si, the dependence of the Bjerrum–Guggenheim osmotic coefficient on magnesium activity is characterized by strong interactions between unlike atoms, manifested as negative deviations from Raoult’s law. Consequently, active interactions occur between magnesium and silicon atoms in the homogeneous melt, leading to the formation of the congruent compound Mg2 Si. For the partial systems Mg2 Si–Mg and Mg2Si–Si, linear correlation relationships with high correlation coefficients were obtained. This indicates that, in the melt along the liquidus line, only van der Waals interaction forces predominate between the elementary structural units of the component under consideration (Mg2Si) and the second component (Mg or Si). Keywords: phase diagrams, liquidus and solidus, Bjerrum-Guggenheim coefficient, mathematical model, activity.