Non-model Calculation of Fusibility Diagrams of Quasi-simple Systems on the example of Electrolyte and Non-electrolyte Systems
Article Sidebar
Main Article Content
Abstract
The article presents a non-model algorithm for calculating the fusibility diagrams of multicomponent systems exclusively from data on the fusibility diagrams of binary subsystems. The first geometric calculation algorithm is based on solving systems of linear equations of liquidus isotherms. The second, thermodynamic calculation algorithm requires only the coordinates of the binary eutectic. The application of both algorithms is demonstrated by the example of fusibility diagrams of ternary quasi-simple salt systems with a common cation and a common anion. There is a convincing agreement between the calculation results the results of the non-model calculation with the available experimental literature data. The proof of the fact that all non-invariant points of various quasi-simple ternary systems (with two identical and one different component) belong to one mono-variant curve is given. Examples of calculations of these mono-variant curves are given and convincing agreement of the non-model calculation with experimental data is demonstrated. A system of transcendental equations has been compiled for calculating the coordinates of ternary eutectic.
Article Details
Copyright (c) 2026 Charykov NA, et al.
This work is licensed under a Creative Commons Attribution 4.0 International License.
Zdanovskiy AB. Patterns in changing the properties of mixed solutions. Proceedings of the Salt Laboratory, Academy of Sciences of the USSR. 1936;(6):70 p.
Ryazanov MA. Selected Chapters of the Theory of Solutions. Syktyvkar: Syktyvkar State University; 1997. 205 p.
Mikulin GI. Thermodynamics of Mixed Solutions of Strong Electrolytes. Leningrad: Chemistry; 1968. p. 202–231.
Charykova MV, Charykov NA. Thermodynamic Modeling of Evaporative Sedimentation Processes. St Petersburg: Nauka; 2003. 261 p. Available from: https://www.researchgate.net/publication/335623004_Thermodynamic_modeling_of_evaporation_processes_of_lunar_and_meteorite_substance
Filippov VK. Systems obeying Zdanovskiy’s rule. Bulletin of Leningrad State University. Physics and Chemistry Series. 1977;(3):98–105.
Wang Zh. The linear concentration rules at constant partial molar quantity Ψ₀ – extension of the Turkdogan and Zdanovskii rules. Acta Metallurgica Sinica. 1980;16(2):195–206. (In Chinese). Available from: https://www.ams.org.cn/EN/Y1980/V16/I2/195
Charykov NA, Rumyantsev AV, Keskinov VA, Letenko DG, Charykova MV, Keskinova MV. Algorithm (procedure variants) for calculation of solubility diagrams of quasi-simple multicomponent water-electrolyte systems. J Chem Eng Data. 2024;69:4089–4097.Available from: https://doi.org/10.1021/acs.jced.4c00307
Charykov NA, Rumyantsev AV, Keskinov VA, et al. A universal algorithm for calculation of vapor–liquid equilibrium diagrams in quasi-simple multicomponent systems. Condensed Matter and Interphases. 2025;27(1):67–85. Available from: https://doi.org/10.17308/kcmf.2025.27/12491
Storonkin AV, Potemin SS. Issues of thermodynamics of eutectic and peritectic systems. In: Issues of Thermodynamics of Heterogeneous Systems and Theory of Surface Phenomena. Issue 4. Leningrad: Leningrad State University; 1977. p. 85–137.
Storonkin AV, Potemin SS. Thermodynamic conditions of pseudo-ideality of ternary heterogeneous systems. In: Issues of Thermodynamics of Heterogeneous Systems and Theory of Surface Phenomena. Issue 7. Leningrad: Leningrad State University; 1985. p. 53–70.
Storonkin AV. Thermodynamics of Heterogeneous Systems. Book 1: Parts I–II. Leningrad: Leningrad State University; 1967. 467 p. Available from: https://www.researchgate.net/publication/226120379_Thermodynamics_of_heterogeneous_systems_with_chemical_interaction
Stonorin AV. Thermodynamics of Heterogeneous Systems. Book 2: Part III. Leningrad: Leningrad State University; 1969. 270 p. Available from: https://www.researchgate.net/publication/226120379_Thermodynamics_of_heterogeneous_systems_with_chemical_interaction
Kogan VB, Ogorodnikov SK, Kafarov VV. Handbook of Solubility. Vol. 3: Ternary and Multicomponent Systems Formed by Inorganic Substances. Book 2. Leningrad: Nauka; 1969. 1171 p.
Prausnitz JM, Lichtenthaler RN, De Azevedo EG. Molecular Thermodynamics of Fluid-Phase Equilibria. 3rd ed. New Jersey: Prentice Hall PTR; 1999. 860 p. Available from: https://www.psgraw.com/wp-content/uploads/2022/05/Molecular-Thermodynamics-of-Fluid-Phase-equilibria-3Ed.pdf
Storonkin AV, Markuzin NP. Investigation of vapor pressure of saturated and unsaturated solutions of potassium chloride in hydrochloric acid. Zhurnal Fizicheskoy Khimii. 1955;29(1):111–119. (In Russian).
Münster A. Chemical Thermodynamics. Moscow: Nauka; 1971. 340 p.
Charykov NA, Rumyantsev AV, Semenov KN, et al. Topological isomorphism of liquid–vapor, fusibility, and solubility diagrams: analogues of Gibbs–Konovalov and Gibbs–Rooseboom laws for solubility diagrams. Processes. 2023;11(5):1405. Available from: https://doi.org/10.3390/pr11051405
Charykov NA, Semenov KN, Keskinov VA. Non-variant phenomena in heterogeneous systems: new type of solubility diagrams points. Ann Adv Chem. 2023;7(1):74–97. Available from: https://doi.org/10.3390/pr1105140510.29328/journal.aac.1001047
Filippov VK, Sokolov VA. Thermodynamics of Heterogeneous Systems and Theory of Surface Phenomena. Vol. 8. Leningrad: Leningrad State University; 1988. p. 39. (In Russian)
Korjinskiy AD. Theoretical Basis of the Analysis of Mineral Paragenesis. Moscow: Nauka; 1973. (In Russia)
Pitzer KS. Thermodynamics of electrolytes. I. Theore tical basis and general equations. J Phys Chem. 1973;77(2):268–277. Available from: https://doi.org/10.1021/j100621a026
Pitzer KS, Kim JJ. Thermodynamics of electrolytes IV. Activity and osmotic coefficients for mixed electrolytes. J Am Chem Soc. 1974;96(18):5701–5707. Available from: https://doi.org/10.1021/ja00825a004
Filippov VK, Charykov NA. Phase equilibria in the Na,Co // Cl,SO₄–H₂O system at 25°C. Russ J Appl Chem. 1986;59(11):2448–2453. Available from: https://www.researchgate.net/publication/293078200_Phase_equilibria_in_the_NaKMgCaSO4Cl-H2O_system_at_25C_in_the_phase_field_of_polyhalite
Filippov VK, Charykov NA. Phase equilibria in the Na,Ni // Cl,SO₄–H₂O system at 25°C. Russ J Inorg Chem. 1988;33(5):1326–1330. Available from: https://www.researchgate.net/publication/357360376_Phase_Equilibria_in_the_Quinary_Na_MgCl_SO4_NO3-H2O_System_at_25C
Mikhailova MP, Litvak AM, Charykov NA, Yakovlev YP. [Title unclear]. Russ Phys Tech Semiconductors. 1997;31(4):410–415. Available from: https://www.researchgate.net/publication/332578578_Discovery_of_III-V_Semiconductors_Physical_Properties_and_Application
Litvak AM, Charykov NA. A new thermodynamic method for calculating melt–solid phase equilibria (ASWU systems). Zhurnal Fizicheskoy Khimii. 1990;64(9):2331–2335.
Litvak AM, Charykov NA. A new thermodynamic method for calculating phase diagrams of binary and ternary systems containing In, Ga, As, and Sb atoms. Reports of the Academy of Sciences of the USSR. Inorganic Materials Series. 1991;27(2):225–230.
Storonkin AV, Vasilkova IV. Questions of the thermodynamics of ternary eutectic and peritectic systems. Issues of Thermodynamics of Heterogeneous Systems and Theory of Surface Phenomena. 1971;(1):3–51.
Grjotheim K, Halvorsen T, Holm JL. The phase diagram of the system NaF–NaCl and thermodynamic properties of fused mixtures. Acta Chem Scand. 1967;21(8):2300–2301. Available from: https://scispace.com/pdf/the-phase-diagram-of-the-system-naf-nacl-and-thermodynamic-3awnaogcdz.pdf
Voskresenskaya NK, Evseeva NN, Berul SI, Vereschagina IP. Handbook on the Fusibility of Salt Systems. Vol. 1: Binary Systems. Moscow–Leningrad: Academy of Sciences of the USSR; 1961.
Plato W. Z. Phys Chem. 1907;58:350. Cited in ref. 30:742.
Wolters A. Neues Jahrb Mineral Geol Palaeontol Beil Bd. 1910;30:55. Cited in ref. 30:742.
Bukhalova GA. Report of Sector of Physico-Chemical Analysis. 1955;26:138. Cited in ref. 30:743.
Rassonskaya IS, Bergman AG. Reports of the Academy of Sciences of the USSR. 1943;38:238. Cited in ref. 30:743.
Rea RF. J Am Ceram Soc. 1938;21:98. Cited in ref. 30:743.
Le Chatelier H. C R Acad Sci. 1894;118:709. Cited in ref. 30:734.
Sackur O. Z Elektrochem Angew Phys Chem. 1910;16:649. Cited in ref. 30:734.
Sackur O. Z Phys Chem. 1912;78:550. Cited in ref. 30:735.
Amadori M. Atti R Accad Lincei. 1913;22(5):366. Cited in ref. 30:735.
Niggli P. Z Anorg Allg Chem. 1919;106:126. Cited in ref. 30:735.
Jiang Y, Sun Y, Liu M, Bruno F, Li S. Eutectic Na₂CO₃–NaCl salt: A new phase change material for high-temperature thermal storage. Sol Energy Mater Sol Cells. 2016;152:155–160. Available from: https://doi.org/10.1016/j.solmat.2016.04.002
Manoilov KE, Smirnov MN. Works of the All-Union Al–Mg Institute. 1940;22:98. Cited in ref. 30:738.
Schmitz-Dumont O, Heckman I. Z Anorg Chem. 1949;260:49. Cited in ref. 30:738.
Volkov NN, Shvab TF. Reports of Physico-Chemical Research Institute of Irkutsk State University. 1953;2(1):60. Cited in ref. 30:738.
Volkov NN, Bergman AG. Reports of the Academy of Sciences of the USSR. 1942;35:50. Cited in ref. 30:738.
Voskresenskaya NK, Evseeva NN, Berul SI, Vereschagina IP. Handbook on the Fusibility of Salt Systems. Vol. 1. Moscow–Leningrad: Academy of Sciences of the USSR; 1961. Available from: https://archive.org/stream/physicalproperti611janz/physicalproperti611janz_djvu.txt
Volkov NM, Bergman AG. Reports of the Academy of Sciences of the USSR. 1940;27:967. Cited in ref. 46:125.
Barzakovsky VP, Lapin VV, Boikova AI, Kurtseva NN. Diagrams of the State of Silicate Systems. Issue IV: Ternary Oxide Systems. Leningrad: Nauka; 1974. 514 p. Available from: https://apps.dtic.mil/sti/citations/AD0787517
Voskresenskaya NK, Evseeva NN, Berul SI, Vereschagina IP. Handbook on the Fusibility of Salt Systems. Vol. 1. Moscow–Leningrad: Academy of Sciences of the USSR; 1961. Available from: https://archive.org/stream/physicalproperti611janz/physicalproperti611janz_djvu.txt
Grjotheim K, Halvorsen T, Holm JL. Acta Chem Scand. 1967;21(8):2300–2301. Available from: https://actachemscand.ki.ku.dk/doi/10.3891/acta.chem.scand.21-2300
Jiang Y, Sun Y, Liu M, Bruno F, Li S. Sol Energy Mater Sol Cells. 2016;152:155–160. Available from: https://doi.org/10.1016/j.solmat.2016.04.002
Grube G. Z Elektrochem Angew Phys Chem. 1927;33:481. Cited in ref. 30:132.
Bergman AG, Banashek EI. Reports of the Physical-Chemical Analysis Sector. 1953;22:196. Cited in ref. 30:133.
Manoilov KE, Smirnov MN. Works of the All-Union Aluminum-Magnesium Institute. 1942;22:98. Cited in ref. 30:196.
Rea RF. J Am Ceram Soc. 1938;21:98. Cited in ref. 30:196.
Fuseya G, Mori M, Immura H. J Soc Chem Ind Jpn. 1933;36:175. Cited in ref. 30:128.
Bukhalova GA, Bergman AG. Russ J Gen Chem. 1951;21:1570. Cited in ref. 30:128.
Bukhalova GI. Russ J Inorg Chem. 1961;6:2359.
Patnaik P. Handbook of Inorganic Chemicals. New York: McGraw-Hill; 2002. ISBN:0-07-049439-8.
Gerward L, Olsen JS, Steenstrup S, Malinowski M, Osbrink S, Waskowska A. A high-pressure X-ray diffraction study of CaF₂. J Appl Crystallogr. 1992;25:578–581. Available from: https://doi.org/10.1107/S0021889892004096
Hohnke DK, Kaiser SW. Epitaxial PbSe and Pb₁₋ₓSₓSe: growth and electrical properties. J Appl Phys. 1974;45(2):892–897. Available from: https://doi.org/10.1063/1.1663334
Nagornyi HI. Report of the Sector of Physico-Chemical Analysis. 1938;11:291. Cited in ref. 49:126.
Wolters H. Neues Jahrb Mineral Geol Palaeontol Beil Bd. 1910;30:50–127. Cited in ref. 49:126–127.
Rassonskaya IS, Bergman AG. Rep Acad Sci USSR. 1943;48:238. Cited in ref. 49:126.
Babev BD. System NaF–NaCl–NaNO₃. Inorg Mater. 2002;38(1):83–84. https://doi.org/10.1023/A:1013667914658
Bukhalova GI. Sector of Physico-Chemical Analysis. 1955;26:268. Cited in ref. 49:476.
Ichaque M. Bull Soc Chim Fr. 1952;(1–2):127. Cited in ref. 49:247.
Gamataeva BY, Bermurzaeva ZI, Gamataev TSh, Gasanaliev AM, Salpagarova ZI. Phase equilibrium in NaF–NaCl–KVO₃ system. Dagestan State Pedagogical University Bulletin. 2015;(4):6–9.
Bergman AG, Dergunov EP. Rep Acad Sci USSR. 1947;58:1369. Cited in ref. 30:475.
Volkov NN, Tumash GP. News of the Physico-Chemical Institute at Irkutsk University. 1953;2(1):60. Cited in ref. 30:475.
Dergunov EP. Rep Acad Sci USSR. 1941;31:752. Cited in ref. 30:475.
Volkov NN, Shvab TF. News of the Physico-Chemical Institute at Irkutsk University. 1953;2(1):41. Cited in ref. 30:475.
Dergunov EP. Rep Acad Sci USSR. 1947;58:1369. Cited in ref. 30:539.
Zheleznyak AV. Physical-Chemical Properties of Hydrocarbon Systems with Carbon Numbers C5–C10. Abstract of PhD Dissertation. Krasnodar, Russia; 2006. 22 p.
Zheleznyak AV. Physical-Chemical Properties of Hydrocarbon Systems with Carbon Numbers C5–C10. PhD Dissertation. Krasnodar, Russia; 2006. 220 p.
Zheleznyak AV, Martsinkovsky AV, Danilin VN. Investigation of the fusibility diagram of the ternary pentane–hexane–heptane system. Physicochemical Analysis of Properties of Multicomponent Systems. 2006;(4):2.
Azamatov BN, Charykov NA, Kuznetsov VV, Kulenova NA, Sadenova MA, Dogadin DS, Keskinov VA, Keskinova MV. Calculation of melting diagrams of a quasi-simple model of ternary systems: algorithm and thermodynamic justification. Phys Solid State. 2025;67(12):2464–2467. https://doi.org/10.21883/0000000000
Charykov NA, Kulenova NA, Sadenova MA, Kuznetsov VV, Azamatov BN, Dogadkin DS, Charykova MV, Keskinova MV, Keskinov VA. Non-model calculation of fusion diagrams for quasi-simple systems from binary subsystem data. Russ J Inorg Chem. 2026;71(2):243–256. https://doi.org/10.1134/S0036023625601412
Charykov NA, Kulenova NA, Sadenova MA, Azamatov BN, Dogadkin DS, Rumyantsev AV, Charykova MV, Keskinova MV, Keskinov VA. Unification of solubility diagrams of multicomponent quasi-simple systems. J Mol Liq. 2025;438:128614–128627. https://doi.org/10.1016/j.molliq.2025.128614
Charykov NA, Kulenova NA, Sadenova MA, Kuznetsov VV, Azamatov BN, Dogadkin DS, Charykova MV. Unifying solubility diagrams of multicomponent quasi-simple systems. Russ J Phys Chem A. 2025;99(13):3302–3319. https://doi.org/10.1134/S0036024425702802
Calculations, datasheets, CAD blocks, and other resources related to science and its subdisciplines. Available from: https://www.piping-designer.com/disciplines/chemical/chemical-elements/4060-pentane
Pentane. NOAA CAMEO Chemicals. Available from: https://cameochemicals.noaa.gov/chris/PTA.pdf
Hexane. Available from: https://www.piping-designer.com/disciplines/chemical/chemical-elements/4050-hexane
n-Hexane. NOAA CAMEO Chemicals. Available from: https://cameochemicals.noaa.gov/chris/HXA.pdf
Heptane. NOAA CAMEO Chemicals. Available from: https://cameochemicals.noaa.gov/chris/HPT.pdf
Heptane (CID 8900). PubChem Compound Summary. Available from: PubChem Heptane ; Heptane. NIST Chemistry WebBook. Available from: https://webbook.nist.gov/cgi/cbook.cgi?ID=C111659&Mask=E
Parks GS, Huffman HM, Thomas SB. Thermal data on organic compounds. VI. Heat capacities, entropies, and free energies of saturated non-benzenoid hydrocarbons. J Am Chem Soc. 1930;52:1032–1041.
Domalski ES, Hearing ED. Heat capacities and entropies of organic compounds in the condensed phase. J Phys Chem Ref Data. 1996;25(1):1. Available from: https://doi.org/10.1063/1.555985
Mondieig D, Rajabalee F, Metivaud V, Oonk HAJ, Cuevas-Diarte MA. n-Alkane binary molecular alloys. Chem Mater. 2004;16(5):786–798. Available from: https://doi.org/10.1021/cm031169p
Nonane. NOAA CAMEO Chemicals. Available from: cameochemicals.noaa.gov›chris/NAN.pdf
Nonane. Available from: https://www.stenutz.eu/chem/solv6.php?name=nonane
Huffman HM, Parks GS, Barmore M. Thermal data on organic compounds. X. Further studies on heat capacities, entropies, and free energies of hydrocarbons. J Am Chem Soc. 1931;53:3876–3888.
Decane. Available from: stenutz.eu›chem/solv6.php?name=decane