Showing papers on "Gibbs–Duhem equation published in 2008"

Solid-Liquid Phase Behavior of Ternary Mixtures

Abstract: The Gibbs Duhem integration technique is extended to calculate ternary phase diagrams at constant temperature and pressure. The technique is used to calculate solid-liquid-vapor phase equilibria for a system selected to roughly model a mixture of two diastereomeric molecules of similar melting point and diameter immersed in a solvent with a lower melting point and a slightly smaller diameter. The cross-species well-depth and diameter between the two diastereomers are varied to determine their impact on the phase equilibria. We find that when the interspecies well-depth is lowered to less than that of either of the diastereomers, the solid phase separates into two solid solutions and consequently there is a region of three-phase coexistence in the ternary phase diagram. We then calculate ternary phase diagrams at a series of temperatures for one set of molecular parameters. For an equimolar mixture of diastereomers, there is a range of temperature and solvent concentration at which only one of the diastereomers will precipitate, thus effecting a separation of the diastereomers. As the temperature is decreased the purity of the precipitate increases.

Computing Activity Coefficients of Binary Lennard-Jones Mixtures by Gibbs-Duhem Integration

Abstract: We determine activity coefficients of both components of different binary Lennard-Jones mixtures obtained at T* = 2.0 and ρ* = 0.6 by using two different strategies: (i) direct evaluation of the excess chemical potentials of both components applying the potential distribution theorem, and (ii) using the Gibbs-Duhem integration based on the activity coefficients of the counter-component, which was calculated directly from the potential distribution theorem approach. Both methods lead numerically to similar results and therefore offer a route to determine activity coefficients of mixtures with large complex molecules by alternatively calculating chemical potentials of potentially simpler counter-constituents.

The geometry of Gibbs-Duhem-Pfaff thermochemical systems

Abstract: The paper deals with problems concerning simple thermochemical systems dynamics, modelled by Gibbs-Duhem-Pfaff equation. Section 1 analyzes the Gibbs-Duhem-Pfaff equation and the associated nonholonomic hypersurface, consisting in integral manifolds with the dimension at most 3. In Section 2 there are listed 6 simple thermodynamical systems with one state. Section 3 describes 15 simple thermodynamical systems with two states. Section 4 defines 20 simple thermodynamical systems with three states. The simple thermodynamical systems depending on measurable state variables are emphasized.

Formulation of the integral Gibbs energy of crystalline elements versus temperature and mechanical stress

Abstract: Phase equilibria calculations require the knowledge of the whole Gibbs energy functions referred to standard elements. Nevertheless, in mechanical calculations the relative change in free energy, already formulated by Landau is often used for the description of the elastic energy of a solid. In order to have a better consistency between the two approaches, it is now suggested a formulation of the integral change of the Gibbs energy of a crystal under stress in equilibrium with a surrounding medium by generalizing the definition of the Gibbs energy of an isotropic substance under hydrostatic pressure.

A note on the application of the phase rule

Abstract: When an equilibrium state contains degenerate phases, the number of independent phases must be considered in applying Gibbs' Phase Rule. This number can be derived from the rank of the matrix containing the coefficients of the Gibbs – Duhem equations. This provides a quantitative determination of the degrees of freedom even under such conditions of degeneracy.

Calculation of solution-vapor thermodynamic equilibria by the method of minimizing the Gibbs energy

Abstract: An algorithm of direct numerical minimization of the Gibbs energy for calculating the liquid-vapor thermodynamic equilibrium in multicomponent systems is proposed. The method makes it possible to simultaneously determine the distributions of concentrations in the vapor and solution and the vapor mole fractionwhen the temperature, pressure, and initial mixture composition are given. The developed algorithm is distinguished from the existing equilibrium calculation methods by a higher stability and the capability of obtaining solutions in the range of parameters in which the conventional algorithms are not able to produce a physically meaningful result. The Peng-Robinson equation of state is used to simulate the Gibbs energy. The accuracy of the proposed algorithm is illustrated by comparison with experimental data.

Summarising mathematical relationships between the Gibbs energy and other thermodynamic information

Abstract: Publisher Summary This chapter presets a schematic summary in the form of a tree showing which mathematical tools relate the Gibbs energy of a system to the various other items. From single-equilibrium calculations, the method of phase mapping based on the Gibbs–Duhem equation leads to the phase diagrams of three different but interrelated kinds. To obtain quantitative relationships in multi-component systems, two-dimensional sections can be generated using appropriate computer programs. Some models, such as the cell model or the quasi-chemical model for ionic liquids, require a phase internal minimization with respect to composition and subsequent numerical differentiation, with all the inherent disadvantages.