Gibbs-Duhem equations phase equilibrium

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). 

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. 

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... 

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... 

A one-component system may have a maximum of three phases in equilibrium. Therefore, the three possible Gibbs-Duhem equations are... 

If one or more chemical reactions are at equilibrium within the system, we can still set up the set of Gibbs-Duhem equations in terms of the components. On the other hand, we can write them in terms of the species present in each phase. In this case the mole numbers of the species are not all independent, but are subject to the condition of mass balance and to the condition that , vtpt must be equal to zero for each independent chemical reaction. When these conditions are substituted into the Gibbs-Duhem equations in terms of species, the resultant equations are the Gibbs-Duhem equations in terms of components. Again, from a study of such sets of equations we can easily determine the number of degrees of freedom and can determine the mathematical relationships between these degrees of freedom. 

The common characteristics of phase transitions are that the Gibbs energy is continuous. Although the conditions of equilibrium and the continuity of the Gibbs energy demand that the chemical potential must be the same in the two phases at a transition point, the molar entropies and the molar volumes are not. If, then, we have two such phases in equilibrium, we have a set of two Gibbs-Duhem equations, the solution of which gives the Clapeyron equation (Eq. (5.73))... 

Gibbs-Duhem Equation and the Phase Rule at Chemical Equilibrium... 

GIBBS-DUHEM EQUATION AND THE PHASE RULE AT CHEMICAL EQUILIBRIUM... 

In discussing one-phase systems in terms of species, the number D of natural variables was found to be Ns + 2 (where the intensive variables are T and P) and the number F of independent intensive variables was found to be Ns + 1 (Section 3.4). When the pH is specified and the acid dissociations are at equilibrium, a system is described in terms of AT reactants (sums of species), and the number D of natural variables is N + 3 (where the intensive variables are T, P, and pH), as indicated by equation 4.1-18. The number N of reactants may be significantly less than the number Ns of species, so that fewer variables are required to describe the state of the system. When the pH is used as an independent variable, the Gibbs-Duhem equation for the system is... 

The Gibbs-Duhem equations for the two phases at equilibrium can be derived from equations 8.2-3 and 8.2-4 ... 

Consider the A-B binary liquid system in equilibrium with the vapour phase at a constant temperature. Is the composition of the vapour the same as that of the liquid Not necessarily. Let s apply the Gibbs-Duhem equation to the liquid phase. 

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

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. 

In order to calculate the distribution coefficient by Equation 1.29, the activity coefficient Y must be evaluated. The activity coefficients are generally determined from the experimental data and correlated on the basis of thermodynamic phase equilibrium principles. The relationship most often used for this purpose is the Gibbs-Duhem equation (Equation 1.7). At constant temperature and pressure, this equation becomes... 

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

Stability criteria are discussed within the framework of equilibrium thermodynamics. Preliminary information about state functions, Legendre transformations, natural variables for the appropriate thermodynamic potentials, Euler s integral theorem for homogeneous functions, the Gibbs-Duhem equation, and the method of Jacobians is required to make this chapter self-contained. Thermal, mechanical, and chemical stability constitute complete thermodynamic stability. Each type of stability is discussed empirically in terms of a unique thermodynamic state function. The rigorous approach to stability, which invokes energy minimization, confirms the empirical results and reveals that r - -1 conditions must be satisfied if an r-component mixture is homogeneous and does not separate into more than one phase. 

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