The Review. The presented results got by author in the field of analytical and computing thermodynamics of the binary systems for the last 25 years. This review includes following sections. In the first section, transition from description of the equilibrium of the phases in two-phase open binary system to description of the equilibrium of the phases in two-phase closed binary system is shown. It allows to go from problem of global minimization in five-dimension space with provision for restrictions to problem of searching for of the global minimum in 3-dimension space with provision for restrictions. The derivation of the system of equations for equilibrium of two phases in closed binary system is given in the second. The third section, different interpretations (as physical, for as geometric and chemical) for the set of the equations for two phases equilibrium for closed binary system are discussed. Analytical description of invariance for the set of equations of phase equilibrium at transformation of molar Gibbs free energies relatively algebra adding of linear functions on compositions with coefficients, which are any functions of temperature and pressure, is proved in the fourth section. The generalization of the Maxwell’s rule in conjugate coordinates for closed binary systems, which is given the fifth section, has allowed to formulate the new algorithm (U-algorithm) of the calculation of the phase diagrams. The autonomous U-algorithm unlike Newton-method (and its modifications), which is describes in the sixth section, does not require the tasks of the initial approach and possesses the square-law velocity to convergence as Newton - method. Some applications of computer programs, which bases on the U-algorithm, for calculated binary phase diagrams, including solution of indirect problems, are described in the seventh section In the eight section obtained results with comparison published papers are discussed. Finally, in the ninth section, we summarize and discuss in short form of some open problems at present and on which are received only first results.
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