Gibbs-Duhem equation system, application

The Gibbs-Duhem equation is applicable to each phase in any heterogenous system. Thus, if the system has P phases, the P equations of Gibbs-Duhem form a set of simultaneous, independent equations in terms of the temperature, the pressure, and the chemical potentials. The number of degrees of freedom available for the particular systems, no matter how complicated, can be determined by the same methods used to derive the phase rule. However, in addition, a large amount of information can be obtained by the solution of the set of simultaneous equations. 

This equation is extremely important (see Section 5.12 for some applications). It is known as the Gibbs-Duhem equation, and such equations as the Duhem-Margules equation may be derived from it. Since no limitation has been put on the type of system considered in the derivation, this equation must be satisfied for every phase in a heterogenous system. We recognize that the convenient independent variables for this equation are the intensive variables the temperature, the pressure, and the chemical potentials. 

Multivariant systems may also become indifferent under special conditions. In all considerations the systems are to be thought of as closed systems with known mole numbers of each component. We consider here only divariant systems of two components. The system is thus a two-phase system. The two Gibbs-Duhem equations applicable to such a system are... 

The derivatives (dP/dT)S3t and (dxt/dT)sat may be determined experimentally or by solution of the set of Gibbs-Duhem equations applicable to each phase, provided we have sufficient knowledge of the system. If the system is multivariant, a sufficient number of intensive variables—the pressure or mole fractions of the components in one or more phases—must be held constant to make the system univariant. Thus, for a divariant system either the pressure or one mole fraction of one of the phases must be held constant. When the pressure is constant, Equation (9.9) becomes... 

Two methods may be used, in general, to obtain the thermodynamic relations that yield the values of the excess chemical potentials or the values of the derivative of one intensive variable. One method, which may be called an integral method, is based on the condition that the chemical potential of a component is the same in any phase in which the component is present. The second method, which may be called a differential method, is based on the solution of the set of Gibbs-Duhem equations applicable to the particular system under study. The results obtained by the integral method must yield... 

B) We have pointed out that experimental studies are usually arranged so that the system is univariant. The experimental measurements then involve the determination of the values of the dependent intensive variables for chosen values of the one independent variable. Actually, the values of only one dependent variable need be determined, because of the condition that the Gibbs-Duhem equations, applicable to the system at equilibrium, must be... 

Binary systems that have a maximum or minimum in the temperature-composition or pressure-composition curves become indifferent at the composition of the maximum or minimum. The Gibbs-Duhem equations applicable to each phase are Equations (10.124) and (10.125). The compositions of the two phases are equal at the maximum or minimum and, therefore, the solution of these two equations becomes... 

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

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. 

The chemical potentials of the various components in a multicomponent system are interrelated. The relationship for binary compounds, known as the Gibbs-Duhem equation, is developed here. Its applicability and usefulness, however, will only become apparent later in Chap. 7. 

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

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.,... 

Finally we looked at the Gibbs-Duhem equation, and how it is used in binary systems. We will see some applications of this in Chapter 14. 

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