Applications of the Gibbs-Duhem equations

The same choice of using component or species is also met in the study of binary systems of molten salts. We could choose to deal solely with components, but because of the concept of the ionic nature of the melts we choose to express the chemical potential of the component in terms of the chemical potential of the species. Equation (8.202) gives the relation for the chemical potential of the general salt Mv+Av in terms of the chemical potentials of the ions. The discussion given in Section 8.18 results in the more convenient equation for the chemical potential of the salt (Eq. (8.208)) 

Equation (10.122) is the basic equation that would be used in obtaining expressions for the excess chemical potential by means of the condition for phase equilibrium. 

We discuss in the next few sections the applications of the Gibbs-Duhem equations to various phase equilibria. In so doing we obtain expressions for the derivatives of one intensive variable with respect to 

Consider a binary, two-phase system without specifying the specific phases. The two Gibbs-Duhem equations on a molar basis are 

Four examples are used to illustrate the methods that can be used. First we consider the derivative (dT/dx p. At constant pressure, Equations (10.124) and (10.125) become 

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To do so we need only specify a dilution sufficient to assure that the activity of component 1 is proportional to its concentration. Application of the Gibbs-Duhem equation leads to Eq. (30). 

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... 

The activity of the solvent (water) in a solution of pure electrolyte dissolved in water can be computed by application of the Gibbs-Duhem equation ... 

Applications of the Gibbs-Duhem equation and the Gibbs phase rule... 

Application of the Gibbs-Duhem equation to the Nd(N03)3-HN03-H2O system gives... 

Now both summation terms vanish upon application of the Gibbs-Duhem equation to each phase. Thus we have, at equilibrium, that... 

In each case the double-summation term vanishes by application of the Gibbs-Duhem equation (Eq. 8.2-15) to each phase. The net information content of Eqs. 8.8-14 is... 

The theoretical method used in relating the water activity to solute activity in each of these methods is an interesting application of the Gibbs-Duhem equation introduced in Chapter 9 ( 9.2.6). There are several ways of doing this for further details see Liu and Lindsay (1972) and Wood et al. (1984). 

CHEMICAL POTENTIAL OF THE SOLUTE IN A BINARY IDEAL SOLUTION APPLICATION OF THE GIBBS-DUHEM EQUATION... 

Thus, the activity coefficient of component i in solution is determined from knowledge derived from measurements of the vapor fugacity of component i in equilibrium with the liquid solution. If the vapor fugacities of components other than i are known, the vapor fugacity of / can be determined by application of the Gibbs-Duhem equation and, again, the activity coefficient of component i can be found. 

Motomura et al. proposed a method of evaluation of various thermodynamic properties of micellar solutions from the surface tension data in the framework of the pseudophase treatment of micellisation [50, 52-55]. According to these authors, the micellar composition at the CMC can be found from the functional dependence of the CMC on the overall surfactant mole fraction using an analogy to the method proposed by Nguyen et al. [49], The approach of Motomura et al. [50] gives also a possibility to determine the relation between the composition of the surface layer and the micelles. Application of the Gibbs-Duhem equation to the whole... 

If weuse dL/ (5, V) = Td5-pdy, the Gibbs-Duhem equations demand- dT-i-Vdp = 0. However, this application of the Gibbs - Duhem equations is wrong. In fact, a similarity of the system is relying, i.e.,... 

In a solution of an electrolyte the activity coefi cientf of the eolverU can be determined by measiirement of its partial pressure, or from the freezing-point depression or the osmotic pressure. The relevant equations are the same as those developed in 9 6. Provided that values of the activity or osmotic coefficient have been determined over a wide range of concentrations, including some solutions which are very dilute, it is possible to calculate the activity coefficient of the solute in some particular solution by application of the Gibbs-Duhem equation. This procedure, as applied to solutions of nonelectrolytes, was described in 9 7 and 9 8. 

The application of the Gibbs-Duhem equation will be dealt with later. 

Application of the Gibbs-Duhem Equation to Determine Yi, Graphically... 

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