Gibbs-Duhem equation phase

Moreover, using the generalized Gibbs-Duhem equations (A2.1.27) for each of the two one-component phases,... 

Each of the coexisting phases will be governed by a Gibbs-Duhem equation so that... 

This result, known as the Gibbs-Duhem equation, imposes a constraint on how the partial molar properties of any phase may vary with temperature, pressure, and composition. In particular, at constant T and P it represents a simple relation among the Af/ to which measured values of partial properties must conform. 

This equation for the surface excesses is the analog of the Gibbs-Duhem equation for bulk phases. 

Many reactions encountered in extractive metallurgy involve dilute solutions of one or a number of impurities in the metal, and sometimes the slag phase. Dilute solutions of less than a few atomic per cent content of the impurity usually conform to Henry s law, according to which the activity coefficient of the solute can be taken as constant. However in the complex solutions which usually occur in these reactions, the interactions of the solutes with one another and with the solvent metal change the values of the solute activity coefficients. There are some approximate procedures to make the interaction coefficients in multicomponent liquids calculable using data drawn from binary data. The simplest form of this procedure is the use of the equation deduced by Darken (1950), as a solution of the ternary Gibbs-Duhem equation for a regular ternary solution, A-B-S, where A-B is the binary solvent... 

During each phase transition of the type illustrated here, both of the intensive parameters P and T remain constant. Because of the difference in density however, when a certain mass of liquid is converted into vapour, the total volume (extensive parameter) expands. From the Gibbs-Duhem equation (8.8) for one mole of each pure phase,... 

The quantities appearing in Eq. (16.2) are not independent. They are related by a Gibbs-Duhem equation, which is obtained in the same way as in the ordinary thermodynamics of bulk phases integrating with respect to the extensive variables results in Ua —TSa — pVa + 7Aa + E/if Nf. Differentiating and comparing with Eq. (16.2) gives ... 

Consider two phases (call them 1 and 2) that reside together in thermodynamic equilibrium. We can apply the Gibbs-Duhem equation (Equation 4.31) for each of the two phases, 1 and 2. 

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). 

The activity of the water is derived from this expression by use of the Gibbs-Duhem equation. To utilize this equation, the interaction parameters fif ) and BH must be estimated for moleculemolecule, molecule-ion and ion-ion interactions. Again the method of Bromley was used for this purpose. Fugacity coefficienls for the vapor phase were determined by the method of Nakamura et al. (JO). 

Using the data of Fig. 38 and data obtained by attempting to duplicate the run, Allen and her coworkers determined the activity coefficients presented in Fig. 39. The two sets of data arc in quite good agreement except at lower mole fractions of DBO, which correspond to the later phases of a run when contamination became significant. Since the activity coefficients for each of the two species were determined from the data, the consistency of the results can be tested by applying the Gibbs-Duhem equation,... 

A review is presented of techniques for the correlation and prediction of vapor-liquid equilibrium data in systems consisting of two volatile components and a salt dissolved in the liquid phase, and for the testing of such data for thermodynamic consistency. The complex interactions comprising salt effect in systems which in effect consist of a concentrated electrolyte in a mixed solvent composed of two liquid components, one or both of which may be polar, are discussed. The difficulties inherent in their characterization and quantitative treatment are described. Attempts to correlate, predict, and test data for thermodynamic consistency in such systems are reviewed under the following headings correlation at fixed liquid composition, extension to entire liquid composition range, prediction from pure-component properties, use of correlations based on the Gibbs-Duhem equation, and the recent special binary approach. 

In this section, we wish to derive the Gibbs-Duhem equation, the fundamental relationship between the allowed variations dRt of the intensive properties of a homogeneous (singlephase) system. Paradoxically, this relationship (which underlies the entire theory of phase equilibria to be developed in Chapter 7) is discovered by considering the fundamental nature of extensive properties Xu as well as the intrinsic scaling property of the fundamental equation U = U(S, V, n, n2,. .., nc) that derives from the extensive nature of U and its Gibbs-space arguments. 

However, from the general Gibbs-Duhem equation (6.34) for each phase, we can write... 

In this case, (12.68) merely recovers the vector form of the Gibbs-Duhem equation for the single phase (cf. Section 10.3), as we should anticipate. 

The latter fact may be emphasized by considering small samples of molar content n and n3 drawn from the bulk phases in regions far from the interface. The size and shape of these samples need have no relationship to the geometry of the interface—any irregularly shaped specimen of bulk phase will do. For each of these bulk phase samples a and p we have Gibbs-Duhem equations... 

Turning now to adsorption equilibrium, let us apply algebraic methods to a two component 1,2 phase system. From the phase rule there will be two degrees of freedom, but we shall reduce this to one by maintaining the temperature constant. Then for the total system there exists a Gibbs-Duhem equation... 

PI4.1 An azeotrope is a constant-boiling solution in which evaporation causes no change in the composition of the liquid. In other words, the composition of the liquid and gaseous phases must be identical. If the vapors may be assumed to be perfect gases, then the ratio of the two partial pressures is equal to the ratio of the mole fractions in the liquid. Use the Gibbs-Duhem equation to show that, at the azeotrope,... 

This is the Gibbs equation, which is particularly important for understanding phase equilibria. A related expression, called the Gibbs-Duhem equation, states that at equilibrium the change of chemical potential of one component results in the change of the chemical potentials of all other components... 

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. 

The derivation of the phase rule is based upon an elementary theorem of algebra. This theorem states that the number of variables to which arbitrary values can be assigned for any set of variables related by a set of simultaneous, independent equations is equal to the difference between the number of variables and the number of equations. Consider a heterogenous system having P phases and composed of C components. We have one Gibbs-Duhem equation of each phase, so we have the set of equations... 

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

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