The values of the Gibbs function of a system with C components create a 2‑dimensional topological manifold that is piecewise smooth and continuous. Each of the C+2 smooth elements of such a manifold represents the states of a phase within the system. The elements are glued together along the C types of phase transformation lines, which converge to a single point that represents the invariant state of the system (i.e. a state with zero degrees of freedom). Transformation lines, treated as edges, and the smooth elements of the manifold, i.e. faces, constitute a zero-vertex graph that represents the invariant state. This graph is referred to here as the graph-map of the invariant state. The distribution of each component in an invariant state depends on the configuration (distribution) of the phase transformation lines. Because the smoothness and continuity of the manifold makes certain configurations of the lines forbidden, some forms of invariant states are also forbidden, even though they satisfy the Gibbs phase rule. Some academic handbooks do not take this fact into account, and provide forbidden configurations as examples of invariant states. States that only differ in terms of the permutation of two or more of their components will belong to the same class. This study shows that all real graph-maps can be represented by C-vertex graphs with C+2 edges that have an even value of the vertex valence. The number of such graphs, i.e. the number of classes of invariant states, ho(C), is shown to meet the recurrence relation ho(2k+1) = 2*ho(2k) - ho(2k-1), where k = 1, 2, 3, 4. Knowing the number ho(C) for several small values of C allows us to determine the number of invariant states in a thermodynamic system using the above equation, regardless of the complexity of the system.
| Published in | American Journal of Physical Chemistry (Volume 8, Issue 1) |
| DOI | 10.11648/j.ajpc.20190801.13 |
| Page(s) | 17-25 |
| Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
| Copyright |
Copyright © The Author(s), 2019. Published by Science Publishing Group |
Graph Theory, Thermodynamic Equilibrium, Invariant Thermodynamic State
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APA Style
Jan Turulski. (2019). Number of Classes of Invariant Equilibrium States in Complex Thermodynamic Systems. American Journal of Physical Chemistry, 8(1), 17-25. https://doi.org/10.11648/j.ajpc.20190801.13
ACS Style
Jan Turulski. Number of Classes of Invariant Equilibrium States in Complex Thermodynamic Systems. Am. J. Phys. Chem. 2019, 8(1), 17-25. doi: 10.11648/j.ajpc.20190801.13
AMA Style
Jan Turulski. Number of Classes of Invariant Equilibrium States in Complex Thermodynamic Systems. Am J Phys Chem. 2019;8(1):17-25. doi: 10.11648/j.ajpc.20190801.13
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Download@article{10.11648/j.ajpc.20190801.13,
author = {Jan Turulski},
title = {Number of Classes of Invariant Equilibrium States in Complex Thermodynamic Systems},
journal = {American Journal of Physical Chemistry},
volume = {8},
number = {1},
pages = {17-25},
doi = {10.11648/j.ajpc.20190801.13},
url = {https://doi.org/10.11648/j.ajpc.20190801.13},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajpc.20190801.13},
abstract = {The values of the Gibbs function of a system with C components create a 2‑dimensional topological manifold that is piecewise smooth and continuous. Each of the C+2 smooth elements of such a manifold represents the states of a phase within the system. The elements are glued together along the C types of phase transformation lines, which converge to a single point that represents the invariant state of the system (i.e. a state with zero degrees of freedom). Transformation lines, treated as edges, and the smooth elements of the manifold, i.e. faces, constitute a zero-vertex graph that represents the invariant state. This graph is referred to here as the graph-map of the invariant state. The distribution of each component in an invariant state depends on the configuration (distribution) of the phase transformation lines. Because the smoothness and continuity of the manifold makes certain configurations of the lines forbidden, some forms of invariant states are also forbidden, even though they satisfy the Gibbs phase rule. Some academic handbooks do not take this fact into account, and provide forbidden configurations as examples of invariant states. States that only differ in terms of the permutation of two or more of their components will belong to the same class. This study shows that all real graph-maps can be represented by C-vertex graphs with C+2 edges that have an even value of the vertex valence. The number of such graphs, i.e. the number of classes of invariant states, ho(C), is shown to meet the recurrence relation ho(2k+1) = 2*ho(2k) - ho(2k-1), where k = 1, 2, 3, 4. Knowing the number ho(C) for several small values of C allows us to determine the number of invariant states in a thermodynamic system using the above equation, regardless of the complexity of the system.},
year = {2019}
}
TY - JOUR T1 - Number of Classes of Invariant Equilibrium States in Complex Thermodynamic Systems AU - Jan Turulski Y1 - 2019/06/11 PY - 2019 N1 - https://doi.org/10.11648/j.ajpc.20190801.13 DO - 10.11648/j.ajpc.20190801.13 T2 - American Journal of Physical Chemistry JF - American Journal of Physical Chemistry JO - American Journal of Physical Chemistry SP - 17 EP - 25 PB - Science Publishing Group SN - 2327-2449 UR - https://doi.org/10.11648/j.ajpc.20190801.13 AB - The values of the Gibbs function of a system with C components create a 2‑dimensional topological manifold that is piecewise smooth and continuous. Each of the C+2 smooth elements of such a manifold represents the states of a phase within the system. The elements are glued together along the C types of phase transformation lines, which converge to a single point that represents the invariant state of the system (i.e. a state with zero degrees of freedom). Transformation lines, treated as edges, and the smooth elements of the manifold, i.e. faces, constitute a zero-vertex graph that represents the invariant state. This graph is referred to here as the graph-map of the invariant state. The distribution of each component in an invariant state depends on the configuration (distribution) of the phase transformation lines. Because the smoothness and continuity of the manifold makes certain configurations of the lines forbidden, some forms of invariant states are also forbidden, even though they satisfy the Gibbs phase rule. Some academic handbooks do not take this fact into account, and provide forbidden configurations as examples of invariant states. States that only differ in terms of the permutation of two or more of their components will belong to the same class. This study shows that all real graph-maps can be represented by C-vertex graphs with C+2 edges that have an even value of the vertex valence. The number of such graphs, i.e. the number of classes of invariant states, ho(C), is shown to meet the recurrence relation ho(2k+1) = 2*ho(2k) - ho(2k-1), where k = 1, 2, 3, 4. Knowing the number ho(C) for several small values of C allows us to determine the number of invariant states in a thermodynamic system using the above equation, regardless of the complexity of the system. VL - 8 IS - 1 ER -
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