Calculation of constrained equilibria by Gibbs energy minimization

Pertti Koukkari (Corresponding Author), Risto Pajarre

Research output: Contribution to journalArticleScientificpeer-review

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Abstract

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
Volume30
Issue number1
DOIs
Publication statusPublished - 2006
MoE publication typeA1 Journal article-refereed

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Gibbs free energy
Chemical reactions
Chemical potential
Reaction kinetics
Interfacial energy
Reaction rates
Pulp
Metal ions
Surface tension
Suspensions
Thermodynamics

Keywords

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

Cite this

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title = "Calculation of constrained equilibria by Gibbs energy minimization",
abstract = "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.",
keywords = "Gibbs energy minimization, thermodynamics, chemical thermodynamics, Lagrange multipliers",
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language = "English",
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pages = "18--26",
journal = "Calphad: Computer Coupling of Phase Diagrams and Thermochemistry",
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Calculation of constrained equilibria by Gibbs energy minimization. / Koukkari, Pertti (Corresponding Author); Pajarre, Risto.

In: Calphad: Computer Coupling of Phase Diagrams and Thermochemistry, Vol. 30, No. 1, 2006, p. 18-26.

Research output: Contribution to journalArticleScientificpeer-review

TY - JOUR

T1 - Calculation of constrained equilibria by Gibbs energy minimization

AU - Koukkari, Pertti

AU - Pajarre, Risto

PY - 2006

Y1 - 2006

N2 - 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.

AB - 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.

KW - Gibbs energy minimization

KW - thermodynamics

KW - chemical thermodynamics

KW - Lagrange multipliers

U2 - 10.1016/j.calphad.2005.11.007

DO - 10.1016/j.calphad.2005.11.007

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JO - Calphad: Computer Coupling of Phase Diagrams and Thermochemistry

JF - Calphad: Computer Coupling of Phase Diagrams and Thermochemistry

SN - 0364-5916

IS - 1

ER -