The Gibbs-Duhem Relation

The argument required here is, of course, the equivalent of the Gibbs-Duhem relation. 

The fundamental thermodynamic equation relating activity coefficients and composition is the Gibbs-Duhem relation which may be expressed as ... 

Taking the partial derivative with respect to the mole fractions in the micelle (x ) and using the Gibbs-Duhem relation to eliminate some of the resulting terms gives... 

It is known, however, from the Gibbs-Duhem relation that... 

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

Assumption of local equilibrium permits the Gibbs-Duhem relation to be written... 

Under the assumption of local equilibrium, the Gibbs-Duhem relation applies, which places an additional constraint on chemical potential changes in Eq. 6.7 and implies that only IV — 1 of the m can vary independently ... 

In addition, because of the Gibbs-Duhem relation, c a dp a + cb d/xs = 0, the chemical potential gradients are interdependent ... 

Using all these variables the relations, which form the starting point for the further calculations, can be constructed. These relations are the energy density , the dissipation function R, the Gibbs-relation and the Gibbs-Duhem relation. To illustrate the idea of our model we split up e and R into several parts according to the different origin of the variables ... 

In a binary solution, the Gibbs-Duhem relation [Eq. (15)] determines the variation of a partial molar property of one component in terms of the variation of the partial molar quantity of the other component. This relation is useful for obtaining chemical potentials in binary solutions when only one of the components has a measurable vapor pressure. Applying Eq. (15) to chemical potentials in a binary solution,... 

Show that the chemical potentials for the ideal solution satisfy the Gibbs-Duhem relation [Eq. (29)]. 

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. 

Gibbs s analytical proof of the adsorption formula is, mutatis mutandis, analogous to the analytical deduction of the Gibbs-Duhem relation both depend on the integration of the formula for the increment in energy, followed by differentiation and comparison of the result with the original formula. 

After applying the Gibbs-Duhem relations to the differential of this equation one is left with ... 

Equation (1.113) is called the Gibbs-Duhem relation, which becomes particularly useful at isobaric and isothermal conditions, and when the force and electrical work are neglected, we have... 

Using the molar-specific volume (v = I IN) and molar-specific entropy (5 = SIN), a simplified version of the Gibbs-Duhem relation results... 

For a binary vapor-liquid system, the Gibbs-Duhem relations are... 

Thermodynamic correction factor Y is defined using the Gibbs-Duhem relation... 

For a system in mechanical equilibrium in which the pressure gradient is balanced by the mass forces, the Gibbs-Duhem relation becomes... 

We now attend to the third term above on the right, beginning with the Gibbs-Duhem relation at constant T and P Equation (1.22.26) reads... 

Examine Fig. 2.3.7 which represents an isotherm obtained from the van der Waals equation, (a) Determine p - fiA from the Gibbs—Duhem relation analytically at fixed temperature, where (iA is the chemical potential at point A in Fig. 2.3.7. (b) Sketch a plot of fi versus P, as obtained from... 

By using Raoult s Law for component 1 and the Gibbs-Duhem relation, show that component 2 must satisfy Henry s Law over the composition range x2 - 1 — xx for which Raoult s Law applies for component 1. (See S. Glasstone, "Thermodynamics for Chemists", D. Van Nostrand, New York, 1947, Chapter 14.)... 

Equation 6.7 is one of the Gibbs-Duhem relations. Substituting for dH from Eq. 6.6 in the equation obtained by differentiating both sides of Eq. 6.5, we can show that... 

When dealing with surfaces, the Gibbs-Duhem relation is an important equation of chemical thermodynamics. It can be derived in the following way [24.25.16.26], The differential of the internal surface energy. dUn. can be expressed as... 

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