Clapeyron and Ehrenfest Equations and Hyper-free Energy for Partly Open Systems

  • Pavel Holba
  • David SedmidubskýEmail author
Part of the Hot Topics in Thermal Analysis and Calorimetry book series (HTTC, volume 11)


Following the approach applied by Clapeyron to describe sharp phase transitions in P-T diagrams of unary systems as well as that used by Ehrenfest for so-called second-order phase transitions, we derived a set of analogous equations for partly open binary and higher-order systems. These systems share one or more components with the surroundings (reservoir), and thus, their content in the system is given by their chemical potentials (activities, a f) in the reservoir. Hence, in addition to P-T diagrams, the phase relations can be represented in Ta f, Pa f, and a fa g phase diagrams and three additional Clapeyronian equations describe the corresponding borderlines delimiting the different phase fields. Moreover, it is shown that Ehrenfest equations cannot be applied for λ-transitions; however, their applicability is demonstrated for so-called partial phase transitions such as liquidus curves in closed binary systems. For partly open systems, 28 new Ehrenfestian equations are derived for partial phase transitions which involve, apart from the changes of heat capacity, thermal expansion and compressibility appearing in the original three Ehrenfest equations, the changes of newly defined quantities such as thermal, pressure, proper, and mutual plutabilities. The Clapeyronian and Ehrenfestian equations derived in this chapter can be useful for equilibrium studies and construction of thermodynamic models of nonstoichiometric phases as well as for the construction of simple phase diagrams reflecting the equilibrium phase relations under a given controlled atmosphere.


Gibbs Free Energy Partial Transition Clapeyronian Equation Free Component Nonstoichiometric Phase 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



P. Holba acknowledges the support of Ministry of Education of the Czech Republic in the framework of CENTEM PLUS project (LO1402) operated under the “National Sustainability Programme I.”


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© Springer International Publishing Switzerland 2017

Authors and Affiliations

  1. 1.New Technologies Research CentreUniversity of West BohemiaPragueCzech Republic
  2. 2.Department of Inorganic ChemistryUniversity of Chemistry and Technology PraguePilsen-PragueCzech Republic

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