Gibbs-Duhem equation solids

In experimental investigations of thermodynamic properties of solutions, it is common that one obtains the activity of only one of the components. This is in particular the case when one of the components constitutes nearly the complete vapour above a solid or liquid solution. A second example is when the activity of one of the components is measured by an electrochemical method. In these cases we can use the Gibbs-Duhem equation to find the activity of the second component. 

By examining the compositional dependence of the equilibrium constant, the provisional thermodynamic properties of the solid solutions can be determined. Activity coefficients for solid phase components may be derived from an application of the Gibbs-Duhem equation to the measured compositional dependence of the equilibrium constant in binary solid solutions (10). 

Most thermodynamic data for solid solutions derived from relatively low-temperature solubility (equilibration) studies have depended on the assumption that equilibrium was experimentally established. Thorstenson and Plummer (10) pointed out that if the experimental data are at equilibrium they are also at stoichiometric saturation. Therefore, through an application of the Gibbs-Duhem equation to the compositional dependence of the equilibrium constant, it is possible to determine independently if equilibrium has been established. No other compositional property of experimental solid solution-aqueous solution equilibria provides an independent test for equilibrium. If equilibrium is demonstrated, the thermodynamic properties of the solid solution are also... 

As a first example, consider a pure liquid in equilibrium with its vapor. Because I wish to focus attention on the liquid/gas interface to the exclusion of adsorption effects at solid boundaries, I shall suppose the containing vessel to be chemically inert. The Gibbs-Duhem equation for the system is then... 

Equation (8.10) can be further simplified by the Gibbs-Duhem equation for the solid solution (A, B)0. As a result, the differential equation for the steady slate demixing profile is found to be... 

II. BASIC THERMODYNAMICS OF SOLID SURFACES A. The Gibbs-Duhem Equation... 

This paper is devoted to the verification of the quality of experimental data regarding the solubility of sparingly soluble solids, such as drugs, environmentally important substances, etc. in mixed solvents. A thermodynamic consistency test based on the Gibbs-Duhem equation for ternary mixtures is suggested. This test has the form of an equation, which connects the solubilities of the solid, and the activity coefficients of the constituents of the solute-free mixed solvent in two mixed solvents of close compositions. 

Two limitations are involved in the derivation of the above equation (1) the compositions of mixed solvents (points c and d) should be close enough to each other for the trapezoidal mle used to integrate the Gibbs-Duhem equation to be valid, (2) the solubility of the solid should be low enough for the activity coefficients of the solvent and cosolvent to be taken equal to those in a solute-free binary solvent mixture. In addition, the fugacity of the solid phase in Eq. (4) should remain the same for all mixed solvent compositions considered. 

It is not possible to apply the Gibbs-Duhem equation at constant T and p to this system because of the requirement of equilibrium between the solution and pure solid 1. Substitution of Eq. (11-147) for pi and Eq. (11-11) for P2 into Eq. (11-158) results in... 

Figure 3.5 Molar volumes (solid lines) and partial molar volumes (broken lines) for binary liquid mixtures. Top is for benzene(l)-carbon tetrachloride(2) mixtures at 25°C, 1 atm. Bottom is for water(l)-ethanol(2) mixtures at 20°C, 1 atm. Note that if the partial molar volume of one component in a binary increases, then by the Gibbs-Duhem equation (3.4.13), the partial molar volume of the other component must decrease. Values computed from data in [4].

The classical Gibbs-Duhem equation must be modified to describe theimodynamic properties of solids. If temperature and pressure are kept constant, one obtains by the function of Everett and Couchman ... 

A liquid mixture in equilibrium with this solid phase contains c thermodynamic components i, where c > c. The Gibbs-Duhem equation for the liquid mixture is ... 

If the partial pressure on the solid-matter system is increased by dp(s), the partial pressure of water vapour changes by a certain quantity dp g) to maintain the equilibrium in the system. According to the Gibbs-Duhem equation (5.14), equilibrium can be maintained under isothermal conditions dT = 0), provided that... 

The Gibbs-Duhem equation limits the possible mathematical forms we may choose to represent liquid-phase (or vapor- or solid-phase) activity coefficients. 

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