Duhem equation application

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... 

On eliminating the chemical potential of the first component between the two Gibbs-Duhem equations applicable to the two phases, we obtain... 

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 ... 

In the case of ternary or higher-order mixtures, solution of the Gibbs-Duhem equation is again based on application of the properties of the exact differentials (Lewis and Randall, 1970) ... 

Surfactant Activity in Micellar Systems. The activities or concentrations of individual surfactant monomers in equilibrium with mixed micelles are the most important quantities predicted by micellar thermodynamic models. These variables often dictate practical performance of surfactant solutions. The monomer concentrations in mixed micellar systems have been measured by ultraf i Itration (I.), dialysis (2), a combination of conductivity and specific ion electrode measurements (3), a method using surface tension of mixtures at and above the CMC <4), gel filtration (5), conductivity (6), specific ion electrode measurements (7), NMR <8), chromatograph c separation of surfactants with a hydrophilic substrate (9> and by application of the Bibbs-Duhem equation to CMC data (iO). Surfactant specific electrodes have been used to measure anionic surfactant activities in single surfactant systems (11.12) and might be useful in mixed systems. ... 

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. 

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

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. 

Many other tests, too numerous to discuss individually, have been devised, all of which are based on the Gibbs-Duhem equation. Only one such test, given by Redlich, is discussed here and is applicable to the case in which both components are volatile and in which experimental studies can be made over the entire range of composition. The reference states are chosen to be the pure liquid at the experimental temperature and a constant arbitrary pressure P0. The values of A/iE[T, P0, x] and A f[T, P0, x] will have been calculated from the experimental data. The molar excess Gibbs energy is given by Equation (10.62), from which we conclude that AGE = 0 when Xj = 0 and when xt = 1. Therefore,... 

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... 

These clearly are not the same as the suggested expressions, which are therefore not correct. Note that application of the summability equation to the derived partial-property expressions reproduces the original equation for H. Note further that differentiation of these same expressions yields results that satisfy the Gibbs/Duhem equation, Eq. (11.14), written ... 

Yet a further application of this important Gibbs-Duhem equation is in the discussion of the triple-phase equilibrium ... 

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. 

This is the Gibbs-Duhem equation, which relates the variation in temperature, pressure, and chemical potentials of the C components in the solution. Of these C + 2 variables, only C + 1 can vary independently. The Gibbs-Duhem equation has many applications, one of which is providing the basis for developing phase equilibrium relationships. 

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 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. 

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). 

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