Skip to main content
SpringerLink
Log in

Application of variational methods in chemical thermodynamics: a study of the stability of chemical systems

  • Original Paper
  • Published:
Journal of Mathematical Chemistry Aims and scope Submit manuscript

Abstract

The stability of chemical systems is discussed by using elementary concepts of control theory, and Lyapunov stability theory. From these concepts, a mathematically clear definition of irreversible thermodynamic processes and equilibrium states is established. Furthermore, it is shown that the extremum principles of classical thermodynamics can be obtained as natural consequences of the Lyapunov stability theory application in usual physicochemical systems. It is also demonstrated that for the case of chemical systems subject to more general physical conditions, it is possible to find a scalar function dependent of macroscopic parameters that describes the evolution of the system toward the equilibrium state. The aim of this work is to show that by using variational methods one can discuss the stability of many chemical systems.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  1. J.W. Gibbs, The Scientific Papers of J. Williard Gibbs. Part 1. Thermodynamics (Longmans-Green, Madison, 1906)

    Google Scholar 

  2. K. Hutter, Acta Mech. 21, 01 (1977)

    Article  Google Scholar 

  3. B.D. Coleman, V.J. Mizel, Arch. Rat. Mech. Anal. 25, 243 (1967)

    CAS  Google Scholar 

  4. J.L. Massera, Ann. Math. 50, 705 (1949)

    Article  Google Scholar 

  5. J.L. Massera, Ann. Math. 64, 182 (1956)

    Article  Google Scholar 

  6. M.C. Reis, Y. Wang, A.B.M.S. Bassi, J. Math. Chem. 52, 441 (2014)

    Article  CAS  Google Scholar 

  7. C.A. Truesdell, The Tragicomical History of Thermodynamics (Springer, Berlin, 1980)

    Google Scholar 

  8. I. Müller, A History of Thermodynamics. The Doctrine of Energy and Entropy (Springer, Berlin, 2007)

    Google Scholar 

  9. R. Giles, Mathematical Foundations of Thermodynamics (Pergamon, Oxford, 1964)

    Google Scholar 

  10. J.C. Willems, Arch. Rat. Mech. Anal. 45, 321 (1972)

    Article  Google Scholar 

  11. J.C. Willems, Eur. J. Control 13, 134 (2007)

    Article  Google Scholar 

  12. R.J.E. Clausius, Ann. Phys. 167, 601 (1854)

    Article  Google Scholar 

  13. W. Thomson, T. Roy, Soc. Edin-Earth 20, 475 (1853)

    Google Scholar 

  14. S.J. Gustafson, I.M. Sigal, Mathematical Concepts of Quantum Mechanics (Springer, Berlin, 2011)

    Book  Google Scholar 

  15. M. Šilhavý, The Mechanics and Thermodynamics of Continuous Media (Springer, Berlin, 1997)

    Google Scholar 

  16. M.E. Gurtin, Arch. Rat. Mech. Anal. 59, 63 (1975)

    Article  Google Scholar 

  17. B.D. Coleman, V.J. Mizel, Arch. Rat. Mech. Anal. 29, 105 (1968)

    Article  Google Scholar 

  18. A. Bacciotti, L. Rosier, Liapunov Functions and Stability in Control Theory, 2nd edn. (Springer, Berlin, 2005)

    Book  Google Scholar 

  19. M.C. Reis, A.B.M.S. Bassi, in Progress in Turbulence V, ed. by A. Talamelli, J. Peinke, M. Oberlack (Springer, Berlin, 2014), p. 195

    Chapter  Google Scholar 

  20. C.A. Truesdell, Rational Thermodynamics, 2nd edn. (Springer, Berlin, 1984)

    Book  Google Scholar 

  21. I.-S. Liu, Continuum Mechanics (Springer, Berlin, 2002)

    Book  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Physical Chemistry, Institute of Chemistry, University of Campinas-UNICAMP, Campinas, SP, Brazil

    Martina Costa Reis & Adalberto Bono Maurizio Sacchi Bassi

Authors
  1. Martina Costa Reis
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Adalberto Bono Maurizio Sacchi Bassi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Martina Costa Reis.

Additional information

The first author acknowledges financial support from the São Paulo Research Foundation (Grant 2013/20872-2).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Reis, M.C., Bassi, A.B.M.S. Application of variational methods in chemical thermodynamics: a study of the stability of chemical systems. J Math Chem 53, 1380–1392 (2015). https://doi.org/10.1007/s10910-015-0494-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10910-015-0494-6

Keywords

Search

Navigation