Showing papers on "Gibbs–Duhem equation published in 2006"

Nonadditive thermodynamic formalism: Equilibrium and Gibbs measures

Abstract: The nonadditive thermodynamic formalism is a generalization of the classical thermodynamic formalism, in which the topological pressure of a single function $\phi$ is replaced by the topological pressure of a sequence of functions $\Phi=(\phi_n)_n$. The theory also includes a variational principle for the topological pressure, although with restrictive assumptions on $\Phi$. Our main objective is to provide a new class of sequences, the so-called almost additive sequences, for which it is possible not only to establish a variational principle, but also to discuss the existence and uniqueness of equilibrium and Gibbs measures. In addition, we give several characterizations of the invariant Gibbs measures, also in terms of an averaging procedure over the periodic points.

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.

Thermodynamic consistency test for high pressure gas–solid solubility data of binary mixtures using genetic algorithms

Abstract: A new computer method to test the thermodynamic consistency of incomplete phase equilibrium data in binary mixtures containing a solid solute and a supercritical fluid, is presented The method is specially designed for treating solubility data that do not cover the whole range of concentration of the components in the mixture The method is based on the Gibbs–Duhem equation and on the appropriate combination between equations of state, mixing rules and combining rules The Peng–Robinson equation of state with the Wong–Sandler mixing rules including the van Laar model for the excess Gibbs free energy required in the mixing rules, are used The model parameters are calculated using genetic algorithms and six gas–solid systems, including 19 isotherms and a total of 362 P–T–y data points were used for the study The systems studied were binary mixtures containing supercritical carbon dioxide with naphthalene, biphenyl, 2,6-dimethylnaphthalene, phenanthrene, anthracene, and pyrene The proposed consistency test method can be used with confidence to determine consistency or inconsistency of a set of experimental solubility data of solids in high pressure gases

Computing the starting state for Gibbs-Duhem integration.

Abstract: Gibbs-Duhem integration implies the numerical integration of a Clapeyron equation. To start the numerical integration, an initial coexistence point and a corresponding initial slope of the Clapeyron equation are needed. In order to apply Gibbs-Duhem integration to all kinds of systems at diverse physical conditions, one has to investigate and assess the available methods that can be used to compute these initial values. This publication focuses on vapor-liquid equilibria in binary mixtures comprising chain molecules. The initial coexistence point is either computed with the NV? Gibbs ensemble or with the Np?+test molecule method with overlapping distributions, which is introduced in this publication. Although computationally demanding, the Np?+test molecule method with overlapping distributions is applicable at conditions where the NV? Gibbs ensemble fails. We investigated three methods that can be employed to compute the initial slope of the Clapeyron equation. The Widom method and the overlapping-distributions difference method provide correct values for the initial slope. The difference method does only provide the correct answer in special cases. The possibility to judge the reliability of the results makes the overlapping-distributions difference method the safest route to the initial slope. Gibbs-Duhem integration requires the frequent computation of the slope of the Clapeyron equation. This slope depends on ensemble averages of the composition. A new bias method for efficient sampling of the composition in a semigrand-canonical simulation of chain molecules is presented. This bias method considerably enhances the composition sampling in systems comprising chain molecules of different sizes.

Thermodynamic properties of liquid Cu–In–Zn alloys

Abstract: The partial Gibbs energies of zinc in the liquid Cu–In–Zn system as a function of temperature and concentration were measured in the temperature range between the melting points of the alloys and 1073K using an EMF method with a liquid electrolyte. The thermodynamic properties were determined for 30 alloys at three different constant ratios of Cu:In of 1:2, 1:1, and 2:1 with zinc added in the concentration range between 5 and 90at.%. The integral Gibbs free energy and the integral enthalpy of mixing of the ternary system were calculated with the Gibbs Duhem equation.

On the Uniqueness of Gibbs States in the Pirogov-Sinai Theory

Abstract: We consider models of classical statistical mechanics satisfying natural stability conditions: a finite spin space, translation-periodic finite potential of finite range, a finite number of ground states meeting Peierls or Gertzik–Pirogov–Sinai condition. The Pirogov–Sinai theory describes the phase diagrams of these models at low temperature regimes. By using the method of doubling and mixing of partition functions we give an alternative elementary proof of the uniqueness of limiting Gibbs states at low temperatures in ground state uniqueness region.

Mass variables and chemical potentials in the Gibbs and De Donder formulations of chemical equilibrium

Abstract: The classical thermodynamic treatment of chemical equilibria in closed systems is founded on theoretical formulations by J.W. Gibbs and Th. De Donder. These two theories represent mathematically equivalent energy variation problems, related to each other through a mass variable transformation analogous to coordinate transformations in mechanics. Associated with the change in mass variables, chemical potentials of reactive substances are defined differently in the two theories, and are subject to different sets of constraints. Traditionally, the two sets of constraints have been merged into one, by assuming that the chemical potential represents the same variable in both theories, an assumption that is formally inconsistent with the difference in chemical potential definitions. Merging of constraints is still possible if chemical potentials remain invariant in value as a manifestation of symmetry under a Gibbs – De Donder mass transformation, but such symmetry has not been investigated. Here we demonstrate chemical potential invariance by reformulating De Donder’s theory using Lagrange multipliers, and combining the resulting constraints on chemical potentials with the classical Gibbs constraints, in a phase equilibration thought experiment. A new form of the Gibbs–Duhem (GD) relation is derived in the De Donder mass system. Comparing this relation to the classical GD equation in Gibbs’ component framework, subject to chemical potential invariance, directly yields the linear operator mapping the De Donder mass variables onto the set of Gibbs component masses. Results are illustrated for a simple ion exchange reaction, in the process solving the longstanding problem of determining solid exchanger activities in ion exchanging mixtures. It is pointed out that, even though the Gibbs and De Donder formulations are mathematically equivalent, the formal structure of De Donder’s theory allows a more explicit and systematic treatment of constraints, and also defines chemical masses in a form more naturally adapted to typical experimental measurements.

Gibbs-Duhem integrations with negative mole fractions

Abstract: A thermodynamic analysis of a molten metal sulfide or oxide system begins with the equilibration of melts with a gas mixture containing the active species, for example, H2S/H2 with sulfides and CO2/CO with oxides. The Gibbs-Duhem equation is then integrated to calculate the activities of the components of interest. Because the sulfur pressures tend toward zero near the metal side of the binary and toward one bar at the metal sulfide side, integration in these systems can be difficult. An alternative approach using negative mole fractions is described and applied to the silver-silver sulfide system.