Journal of Advanced Mathematical Modeling

Journal of Advanced Mathematical Modeling

Graphical Depiction of Concavity and Convexity in the Context of Thermodynamic Stability

Document Type : Original Paper

Author
Mehrdad Bamdad
Chemistry Department,, Faculty of Science,, Shahid Chamran University of Ahvaz,, Ahvaz,, Iran
10.22055/jamm.2026.50167.2388
Abstract
Thermodynamic stability of systems at equilibrium represents a fundamental and crucial topic within the realm of thermodynamics. The stability of thermodynamic systems at equilibrium inherently related to the concavity of the entropy (S) or the convexity of the internal energy (U), the functions those encountered in mathematics. It can be demonstrated that for entropy representation formalism S = S(U, V, N) for a stable system, entropy versus internal energy at constant volume (V) and constant number of particles (N expressed in moles) is characterized as a concave function. As a result when energy is transferred between subsystems, at constant volume and constant number of particles within the subsystems, the entropy does not demonstrate an increase. In other words a closed system (the internal energy, volume and all the mole numbers are constant) at equilibrium reaches a state of maximum entropy. This study has been shown that by using the graphical illustration of both convex and concave functions, and by transforming one function into the other via rotation, the notion of thermodynamic stability within the context of entropy representation can be transformed to the framework of internal energy representation. As by a ninety-degree counterclockwise rotation, a concave function is converted into a convex function, similarly by rotation an S-U graph a U-S graph will be obtained that represent the convexity of internal energy as a function of entropy.
Keywords
Thermodynamic Stability
Concave Upward
Concave Downward
Entropy
Internal Energy
Subjects
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  • Receive Date 14 July 2025
  • Revise Date 15 December 2025
  • Accept Date 26 January 2026