Ternary systems Gibbs-Duhem equation

By means of the substitution y = jc/ /7 , these equations can be converted into ordinary differential equations. An analogous equation could also be written for the change in concentration of iron. However, because of eq. (7-17) and because of thermodynamic relationships in the ternary system (Gibbs-Duhem equation), this equation would not be independent. Because of the higher mobility of carbon, it is observed that the carbon reaches equilibrium after a relatively short time (as compared to the diffusion of silicon). From eq. (7-15) we may write ... 

In the case of the following systems, predict the distribution of substance C, and the selectivity of solvent B for C, using the data indicated and the appropriate ternary integrated Gibbs-Duhem equation. Compare with the observed data reported in the... 

The experimental studies of three-component systems based on phase equilibria follow the same principles and methods discussed for two-component systems. The integral form of the equations remains the same. The added complexity is the additional composition variable the excess chemical potentials become functions of two composition variables, rather than one. Because of the similarity, only those topics that are pertinent to ternary systems are discussed in this section of the chapter. We introduce pseudobinary systems, discuss methods of determining the excess chemical potentials of two of the components from the experimental determination of the excess chemical potential of the third component, apply the set of Gibbs-Duhem equations to only one type of phase equilibria in order to illustrate additional problems that occur in the use of these equations, and finally discuss one additional type of phase equilibria. 

As in Section 4.1, it is convenient to label exchanging ions with numerical indices in describing multicomponent ion exchange equilibria. The thermodynamic approach will be illustrated for a ternary cation exchange system (e.g., Na+, Mg2+, Ca2+), but the extension to an N-ary system is direct. This is evident, for example, in the Gibbs-Duhem equation for a ternary system ... 

The Gibbs-Duhem equation for ternary mixtures is used to analyze the quality of experimental data pertaining to the solubility of drugs and other poorly soluble solids in a binary mixed solvent. In order to test the quality of the data, a thermodynamic consistency test is suggested. This test is based on the thermodynamic relation between the solubilities of a solid in a binary mixed solvent at two different compositions and the activity coefficients of the constituents of the solute-free mixed solvent. The suggested test is applicable to all kinds of systems with the following limitations (1) the solubility of the solid should be low, (2) the above two compositions of the mixed solvent should be close enough to each other. 

Schumann, R. Jr., 1955, Application of Gibbs-Duhem equations to ternary systems Acta Metallurgica, v. 3, pp. 219-226. 

The exact calculation equations are given in [25], where it has also been proved that the Gibbs-Duhem equation is fulfilled. As well, NRTL parameters have been fitted up to molalities of 30mol/kg for a number of systems. Together with the ionic diameters, they are listed in [25]. Osmotic and mean ionic activity coefficients could be reproduced in an excellent way for a number of systems. Furthermore, the parameters fitted to binary systems have been successfully applied to ternary systems, that is, one salt in a binary solvent mixture, which always causes problems with the Electrolyte NRTL model [25]. 

This process can be extended to ternary or even higher order systems, and it can be shown that all chemical potentials in a multicomponent system can be evaluated if the potential of one component is known over the whole compositional space. However, the process becomes complex, and has been little used even in ternary systems. Pitzer and Brewer (1961, Chapter 34) have a discussion of this, with several useful references. The vast majority of uses of the Gibbs-Duhem relation have been in binary systems, using variations of Equation (4.73). We should mention that the term Gibbs-Duhem is commonly applied to any of Equations (4.68)-(4.73). 

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