The Characteristic Curves of Water

  • Arnold Neumaier
  • Ulrich K. Deiters


In 1960, E. H. Brown defined a set of characteristic curves (also known as ideal curves) of pure fluids, along which some thermodynamic properties match those of an ideal gas. These curves are used for testing the extrapolation behaviour of equations of state. This work is revisited, and an elegant representation of the first-order characteristic curves as level curves of a master function is proposed. It is shown that Brown’s postulate—that these curves are unique and dome-shaped in a double-logarithmic pT representation—may fail for fluids exhibiting a density anomaly. A careful study of the Amagat curve (Joule inversion curve) generated from the IAPWS-95 reference equation of state for water reveals the existence of an additional branch.


Brown’s characteristic curve Ideal curve IAPWS-95 equation of state Joule inversion Joule–Thomson inversion Water 



ith virial coefficient


Isobaric heat capacity


Isochoric heat capacity


(Dimensionless) thermodynamic response function, \(X \in \{p, V, T, \kappa \}\) (Eqs. 1415)


Gibbs energy


Hessian of \(G_\mathrm{m}(p, T)\)


Configurational enthalpy


Amount of substance




Dimensionless logarithmic slope, \(X \in \{\mathrm{A, B, C}\}\) (Eqs. 1618)


Universal gas constant


Configurational entropy




Configurational internal energy




Compression factor, \(Z = p V_\mathrm{m}/(R T)\)

\(\alpha _p\)

Isobaric thermal expansivity

\(\kappa _T\)

Isothermal compressibility

\(\rho \)

Molar density, \(\rho = V_\mathrm{m}^{-1}\)



Amagat (Joule inversion) curve


Boyle curve


Critical property


Charles (Joule–Thomson inversion) curve


Molar property



Ideal gas


Vapour phase


Liquid phase


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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ViennaWienAustria
  2. 2.Institute of Physical ChemistryUniversity of CologneKölnGermany

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