Showing papers on "Gibbs–Duhem equation published in 2013"

Molecular simulation of aqueous electrolytes: water chemical potential results and Gibbs-Duhem equation consistency tests.

Abstract: This paper deals with molecular simulation of the chemical potentials in aqueous electrolyte solutions for the water solvent and its relationship to chemical potential simulation results for the electrolyte solute. We use the Gibbs-Duhem equation linking the concentration dependence of these quantities to test the thermodynamic consistency of separate calculations of each quantity. We consider aqueous NaCl solutions at ambient conditions, using the standard SPC/E force field for water and the Joung-Cheatham force field for the electrolyte. We calculate the water chemical potential using the osmotic ensemble Monte Carlo algorithm by varying the number of water molecules at a constant amount of solute. We demonstrate numerical consistency of these results in terms of the Gibbs-Duhem equation in conjunction with our previous calculations of the electrolyte chemical potential. We present the chemical potential vs molality curves for both solvent and solute in the form of appropriately chosen analytical equations fitted to the simulation data. As a byproduct, in the context of the force fields considered, we also obtain values for the Henry convention standard molar chemical potential for aqueous NaCl using molality as the concentration variable and for the chemical potential of pure SPC/E water. These values are in reasonable agreement with the experimental values.

Thermodynamics and Phase Diagrams of Materials

Abstract: The sections in this article are Introduction Notation Gibbs Energy and Equilibrium Gibbs Energy Chemical Equilibrium Predominance Diagrams Calculation of Predominance Diagrams Ellingham Diagrams as Predominance Diagrams Discussion of Predominance Diagrams Thermodynamics of Solutions Gibbs Energy of Mixing Chemical Potential Tangent Construction Gibbs–Duhem Equation Relative Partial Properties Activity Ideal Raoultian Solutions Excess Properties Activity Coefficient Multicomponent Solutions Binary Phase Diagrams Systems with Complete Solid and Liquid Miscibility Thermodynamic Origin of Phase Diagrams Pressure–Composition Phase Diagrams Minima and Maxima in Two-Phase Regions Miscibility Gaps Simple Eutectic Systems Regular Solution Theory Thermodynamic Origin of Simple Phase Diagrams Illustrated by Regular Solution Theory Immiscibility–Monotectics Intermediate Phases Limited Mutual Solubility–Ideal Henrian Solutions Geometry of Binary Phase Diagrams Application of Thermodynamics to Phase Diagram Analysis Thermodynamic/Phase Diagram Optimization Polynomial Representation of Excess Properties Least-Squares Optimization Calculation of Metastable Phase Boundaries Ternary and Multicomponent Phase Diagrams The Ternary Composition Triangle Ternary Space Model Polythermal Projections of Liquidus Surfaces Ternary Isothermal Sections Topology of Ternary Isothermal Sections Ternary Isopleths (Constant Composition Sections) Quasi-Binary Phase Diagrams Multicomponent Phase Diagrams Nomenclature for Invariant Reactions Reciprocal Ternary Phase Diagrams Phase Diagrams with Potentials as Axes General Phase Diagram Geometry General Geometrical Rules for All True Phase Diagram Sections Zero Phase Fraction Lines Choice of Axes and Constants of True Phase Diagrams Tie-lines Corresponding Phase Diagrams Theoretical Considerations Other Sets of Conjugate Pairs Solution Models Sublattice Models All Sublattices Except One Occupied by Only One Species Ionic Solutions Interstitial Solutions Ceramic Solutions The Compound Energy Formalism Non-Stoichiometric Compounds Polymer Solutions Calculation of Limiting Slopes of Phase Boundaries Short-Range Ordering Long-Range Ordering Calculation of Ternary Phase Diagrams From Binary Data Minimization of Gibbs Energy Phase Diagram Calculation Bibliography Phase Diagram Compilations Thermodynamic Compilations General Reading

Long-range interacting systems and the Gibbs-Duhem equation

Abstract: The generalized Gibbs-Duhem equation is obtained for systems with long-range interactions in $d$ spatial dimensions. We consider that particles in the system interact through a slowly decaying pair potential of the form $1/r^ u$ with $0\leq u\leq d$. The local equation of state is obtained by computing the local entropy per particle and using the condition of local thermodynamic equilibrium. This local equation of state turns out to be that of an ideal gas. Integrating the relation satisfied by local thermodynamic variables over the volume, the equation involving global magnitudes is derived. Thus, the Euler relation is found and we show that it is modified by the addition of a term proportional to the total potential energy. This term is responsible for the modification of the Gibbs-Duhem equation. We also point out a close relationship between the thermodynamics of long-range interacting systems and the thermodynamics of small systems introduced by Hill.

Basic Thermodynamics of Electrified Interfaces

Abstract: In this study some general aspects of the thermodynamics of systems with interfaces are discussed, and a brief treatment of interfaces within the framework of classical thermodynamics is presented. Special attention is paid to the theory of electrified interfaces. The intensive parameter conjugate to surface area (“surface tension” or “interfacial tension”) is an important parameter also in the thermodynamic theory of electrodes, because the interactions between the adjacent bulk phases take place via interfaces, for example, via the interface between a metal and an electrolyte solution. As a consequence, the thermodynamic properties of the interface region (i.e., the electronic conductor/ionic conductor interface) directly influence the electrochemical processes. First, to introduce the reader to the topic, basic concepts (such as “surface,” “interface,” “interphase,” “interfacial or interface region,” “dividing surface,” “adsorption”) are reviewed, a reasonably simple thermodynamic treatment of interfaces, together with a brief description of the models widely used in the literature, are presented, and the characteristics of the Gibbs “dividing plane” model and the Guggenheim “interphase” model are outlined. The derivation of the electrocapillary equation, the Gibbs adsorption equation, and the Lippmann equation for an ideally polarizable electrode is given. A simple illustrative example for the application of the electrocapillary equation is presented. Some important mathematical concepts (e.g., theory of homogeneous functions and partly homogeneous functions, Euler's theorem, and Legendre transformation) and various functional relationships of the thermodynamics of surfaces and interfaces are summarized.

The Free Energy and Gibbs Measure

Abstract: In Sect.1.1, we will introduce the Sherrington–Kirkpatrick model and a family of closely related mixed p-spin models and give some motivation for the problem of computing the free energy in these models. A solution of this problem in Chap.?? will be based on a description of the structure of the Gibbs measure in the thermodynamic limit and in this chapter we will outline several connections between the free energy and Gibbs measure. At the same time, we will introduce various ideas and techniques, such as the Gaussian integration by parts, Gaussian interpolation, and Gaussian concentration, that will play essential roles in the key results of this chapter and throughout the book. In the last section, we will prove the Dovbysh–Sudakov representation for Gram-de Finetti arrays, which will allow us to define a certain analogue of the Gibbs measure in the thermodynamic limit. As a first step, we will prove the Aldous–Hoover representation for exchangeable and weakly exchangeable arrays. In Sect.1.4, we will give a classic probabilistic proof of this result for weakly exchangeable arrays and, for a change, in the Appendix we will prove the representation for exchangeable arrays using a different approach, based on more recent ideas of Lovasz and Szegedy in the framework of limits of dense graph sequences. We will describe another application of the Aldous–Hoover representations for exchangeable arrays in Chap.??.