On Selected Properties of the Gibbs Function Topological Manifold

Document Type : Research Paper


Third Age University, 16035 Czarna Wies Koscielna, Poland, and Chemistry Institute, University at Bialystok, 15443, Bialystok, Poland



Quantitatively, the equilibrium in classical thermodynamics in the C-component isobaric-isothermal system is determined by the minimum value of the Gibbs function. The topological manifold of this function is a 2-D dimensional, smooth piece, geometric creation. These pieces represent individual states of single-phase systems. Successive pieces of the manifold are glued along the line of phase transitions to form the manifold of the whole, en bloc, C-component system. Gluing smooth pieces together must guarantee the continuity of the glued whole. The study found the dependence of the number of ways of gluing single-phase pieces on the number of components of the system. It has also been shown that the distribution of components in individual phases of the system is represented by a planar graph with 4 faces, called a normal graph.
Studies of the topological properties of the manifold fragments representing single-phase equilibrium states indicate that the value of the Gibbs potential in these states is encoded in the geometry of the topological manifold. In concrete terms, this value is equal to the length of the minimum path lying on the surface of the manifold, connecting the various degrees of freedom of the system (the vertices of the graph). In complex systems, with very large C, the number of paths connecting the degrees of freedom is monstrously large. Preliminary calculations show that in such systems the number of paths with a minimum length or not much different from it may be greater than one.


  1. O. Rüdel, The phase rule and the boundary law of Euler, Z. Electrochem. 35 (1929) 54-57.
  2. M. A. Klochko, Analogy between phase rule and the Euler theorem for polyhedrons, Izvest. Sectora Fiz.-Khim. Anal., Inst. Obshei. i Neorg. Khim., Akadd. Nauk SSSR 19 (1949) 82-88.
  3. D. R. Rouvray, Uses of graph theory, Chem. Br. 10 (1974) 11-15.
  4. T. P. Radhakrishnan, Euler's formula and phase rule, J. Math. Chem. 5 (1990) 381-387.
  5. L. Pogliani, Phase diagrams and physiochemical graphs. How did it start?, MATCH Commun. Math. Comput. Chem. 49 (2003) 141-152.
  6. A. I. Seifer and V. S. Stein, Topology composition-property of diagram phase, Zh. Neorg. Khim. 6 (1961) 2711-2723.
  7. J. Mindel, Gibbs' phase rule and Euler's formula, J.Chem. Educ. 39 (1962) 512-514.
  8. F. A. Weinhold, Theoretical Chemistry, Advances and Perspectives, vol. 3, (Eds.: H. Eyring, D. Henderson), Academic Press, NY, 1978, pp. 15-54.
  9. J. Turulski and J. Niedzielski, Use of graph theory in thermodynamics of phase equilibria, J. Chem. Inf. Comput. Sci. 42 (2002) 534-539.
  10. N. Trinajstić, Chemical Graph Theory, 2nd Ed., CRC Press, Boca Raton, FI, 1992.
  11. J. Turulski, Dimension of the Gibbs function topological manifold, J. Math. Chem. 53 (2) (2015) 495-513.
  12. J. Turulski, Number of classes of invariant equilibrium states in complex thermodynamic systems, Amer. J. Chem. Phys. 8 (1) (2019) 17-25.
  13. H. DeVoe, Thermodynamics and Chemistry, (November 2021 web version at: https://open.umn.edu/opentextbooks/textbooks/715, pp. 231, 405, 419 .
  14. H. Bukowski and W. Ufnalski, Solutions (in Polish), W. N.-T. Warsaw, 1995, p. 39.
  15. F. Harary and R. C. Read, Is the null-graph a pointless concept? Graphs and Combinatorics, Proc. Capital Conf., George Washington Univ., Washington D.C., 1973), pp. 37-44. Lecture Notes in Math., Vol. 406, Springer, Berlin, 1974.
  16. M. Ch. Karapientiantz, Chemical Thermodynamics (in Russian), Khimia, Moscow, 1975, p.122.
  17. N. J. A. Sloane and S. Plouffe, The Encyclopedia of Integer Sequences, Academic Press, San Diego, 1995.
  18. J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus, Springer-Verlag New York, Inc., 1998.
  19. J. V. Knop, W. R. Müller, K. Szymanski and N. Trinastić, Computer Generation of Certain Classes of Molecules, SKTH/Kemija u Industriji: Zagreb, Croatia, 1985.
  20. A. Cayley, A theorem on trees, Quart. J. Pure Appl. Math. (1889) 376-378.
  21. F. Harary and E. M. Palmer, Graphical Enumeration, (Chapter: Asymptotics), Academic Press, NY, 1973.
  22. J. Turulski and J. Niedzielski, Earth's biodiversity as interpreted by a chemist, J. Math. Chem. 36 (1) (2004) 29-40.
  23. J. Turulski, Binary self-similar chemical structures, J. Math. Chem. 56 (2018) 850–863.
  24. J. Turulski, Dimension of the Gibbs function topological manifold: 2. Thermodynamically stable binary quasicrystals: Reality or artefact?, J. Math. Chem. 53 (2) (2015) 517-526.
  25. A. P. Tsai, J. Q. Guo, E. Abe, H. Takakura and T. J. Sato, Stable binary quasicrystals, Nature 408 (2000) 537-538.
  26. A. I. Goldman, A. Kreyssig, S. Nandi, M. G. Kim, M. L. Caudle and P. C. Canfield, High-energy X-ray diffraction studies of i-Sc12Zn88, Phil. Mag. 91 (19-21) (2011) 2427-2433.
  27. A. I. Goldman, T. Kong, A. Kreyssig, A. Jesche, M. Ramazanoglu, K. W. Dennis, S. L. Bud’ko and P. C. Canfield, A family of binary magnetic icosahedral quasicrystals based on rare earths and cadmium, Nat. Mater. 12 (2013) 714-718.
  28. S. R. Brinkley, Calculation of the equilibrium composition of systems of many constituents, J. Chem. Phys. 15 (1947) 107-110.
  29. M. Zhao, Z. Wang and L. Xiao, Determining the number of independent components by Brinkley's method, J. Chem. Educ. 69 (1992) 539-542.
  30. G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers, (Chapter: 21.9), McGraw Hill Book Company, NY, 1961.