Gibbs Duhem condition

On the other hand, for surfactant micelles or microemulsion droplets that are formed by aggregation in a one-phase region of a phase diagram, there will be one more degree of freedom, in all r + 3 the r+l thermodynamic state variables plus two geometrical variables, in full agreement with the way we have written the Gibbs-Duhem conditions (Eqs. (36), (37), and (40)) above (where the temperature variable is omitted). 

Equation (16) is a differential equation and applies equally to activity coefficients normalized by the symmetric or unsymme-tric convention. It is only in the integrated form of the Gibbs-Duhem equation that the type of normalization enters as a boundary condition. 

We may also note the special form of the Gibbs-Duhem equation (6.34) under laboratory conditions of constant T and P, namely,... 

As usual, our goal is to find the minimum of G( ) in order to determine the equilibrium position ( = eq) of the chemical reaction at constant T and P. From (6.10c) [cf. the Gibbs -Duhem equation (6.36a)], the differential dG under these conditions is simply... 

For the external, observer, jA +/B = 0. From this condition and the Gibbs-Duhem relation, the local lattice velocity becomes... 

Here we have chosen AO, B203, and oxygen O as components of the ternary compound AB204 (or rather (A,B)304+l5). Since nAO+Pb,o3 — U°ab2o, (which is the Gibbs-Duhem equation integrated under the assumption that the spinel is strictly stoichiometric and stress effects can be neglected), we obtain from the cation fluxes and the steady state condition jA/cA = jB/cB = vb... 

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. 

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... 

This equation is very similar to the Gibbs-Duhem equation under the condition that the temperature and pressure are constant. A more general relation can be obtained by differentiating Equation (6.10) and comparing the result with Equation (6.1). The differentiation of Equation (6.10) gives... 

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... 

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))... 

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... 

Thus, another approach consists in selecting some boundary conditions and properties. It is obvious that all exact correlation functions must satisfy and incorporate them in the closure expressions at the outset, so that the resulting correlations and properties are consistent with these criteria. These criteria have to include the class of Zero-Separation Theorems (ZSTs) [71,72] on the cavity function v(r), the indirect correlation function y(r) and the bridge function B(r) at zero separation (r = 0). As will be seen, this concept is necessary to treat various problems for open systems, such as phase equilibria. For example, the calculation of the excess chemical potential fi(iex is much more difficult to achieve than the calculation of usual thermodynamic properties since one of the constraints it has to satisfy is the Gibbs-Duhem relation... 

The advantage of this choice of the X dependence for the correlation functions and the bridge function relies on the fact that the excess chemical potential, and the one-particle bridge function as well, can be determined unambiguously in terms of B(r) as soon as n and m are known. To address this problem, the authors proposed to determine the couple of parameters (n m) in using the Gibbs-Duhem relation. This amounts to obtaining values of n and m from Eq. (87), which is considered as supplementary thermodynamic consistency condition that have to be fulfiled. 

The present relations differ from the KM approximation since the factor 3 is replaced by the bridge function at zero separation. This feature does not seem to be unreasonable because, from diagrammatic expansions, B (r) = B r)/3 is supposed to be accurate only at very low densities. Eq. (112) presents two advantages at high density i) it provides a closed-form expression for Bother fluids than the HS model and ii) it allows to ensure a consistent calculation of the excess chemical potential by requiring only the use of the pressure consistency condition (the Gibbs-Duhem constraint, no longer required, is nevertheless implicitly satisfied within 1%). 

Equations 5.33 and 5.36 constitute four differential equations in the three dependent variables, fj, f2, f3 (the mole fractions and the conditional equilibrium constants are assumed to be known through measurement). If there are more than three exchanging ions, for each term added to the Gibbs-Duhem equation, there will be additional constraints like Eq. 5.36, so that enough equations always will be generated to express the activity coefficients in terms of the conditional equilibrium constants, as in Eq. 5.25. 

Related Calculations. When vapor-liquid equilibrium data are taken under isobaric rather than isothermal conditions, as is often the case, the right-hand side of the preceding Gibbs-Duhem equation cannot as readily be taken to approximate zero. Instead, the equation should be taken as... 

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