Calculation of constrained equilibria by Gibbs energy minimization

Pertti Koukkari (Corresponding Author), Risto Pajarre

Research output: Contribution to journalArticleScientificpeer-review

51 Citations (Scopus)


The Gibbs energy minimization encompasses active use of the chemical potentials (partial molar Gibbs energies) of the constituents of the system. Usually, these appear at their equilibrium values as a result of the minimization calculation, the mass balance constraints being the necessary subsidiary conditions. Yet, there are several such physico-chemical circumstances where the system is also constrained by other factors, such as surface effects, potential fields or even by chemical reaction kinetics. In this paper a particular method is presented by which constrained chemical potentials can be applied in a multi-phase Gibbs energy minimization. The constrained potentials arise typically from work-related thermodynamic displacements in the system. When Gibbs energy minimization is performed by the Lagrange method, these constraints appear as additional Lagrangian multipliers. Examples of the constrained potential method are presented in terms of the electrochemical Donnan equilibria in aqueous systems containing semi-permeable interfaces, the phase formation in surface-energy controlled systems and in systems with affinities controlled by chemical reaction kinetics. The methods have been applied successfully in calculating distribution coefficients for metal ions together with pH-values in pulp suspensions, in the calculation of surface tension of alloys, and in thermochemical process modeling involving chemical reaction rates.
Original languageEnglish
Pages (from-to)18-26
Number of pages9
JournalCalphad: Computer Coupling of Phase Diagrams and Thermochemistry
Issue number1
Publication statusPublished - 2006
MoE publication typeA1 Journal article-refereed


  • Gibbs energy minimization
  • thermodynamics
  • chemical thermodynamics
  • Lagrange multipliers

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