Surface Gibbs-Duhem equation

Often the units for n are dyn m to yield numbers that are simpler. This is not necessary, however. We prefers to leave all units in SI for simplicity. Today n is often reported in units of mJ m ) For most thermodynamic treatments (see Hiemenz [12]), the surface Gibbs-Duhem equation would be... 

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

The surface concentrations T depend on the thickness of the interfacial region, and we would like to express them through quantities which are independent of it. This can be done for those species which occur both at the interface and in the solution. Usually one of the components of the solution, the solvent, has a much higher concentration then the others. We denote it by the index 0 , and introduce surface excesses with respect to the solvent in the following way In the bulk of the solution the Gibbs-Duhem equation (at constant T and p) is simply E Ni dfri = 0, or ... 

Differentiation of eq. (6.77), when combined with eq. (6.14), gives the Gibbs-Duhem equation in internal energy for a system where the surface energy is not negligible ... 

Gibbs approach, for treating surfactant adsorption, 24 134-138 Gibbs dividing surface, 24 135 Gibbs-Duhem equation, 3 744 24 134, 135, 672, 677... 

Following our development of the Gibbs-Duhem equation in Chapter 8, we now apply Eq. (45) to a process at constant T and P, where components are added to the surface in an arbitrary ratio. In this process, the chemical potentials will vary ... 

The Gibbs-Duhem equation is used in several cases in electrochemistry, e.g., in the derivation of - Gibbs adsorption equation or -> Gibbs-Lippmann equation since Eq. (4) can be extended by surface work ... 

II. BASIC THERMODYNAMICS OF SOLID SURFACES A. The Gibbs-Duhem Equation... 

It is instructive to compare this relation with the Gibbs-Duhem equation for a liquid surface. For a liquid, no elastic strain can exist. Consequently dee must be zero and ys = y. This leads to... 

A relation between the surface tension y and the surface stress T can be directly derived from the Gibbs-Duhem equation, Eq. (7). At constant temperature, chemical potentials, and electric potential we have... 

Adsorption is described by the Gibbs-Duhem equation. Considering only a single vapor at constant temperature, electric potential, and elastic strain, the change in surface tension is given by... 

The surface entropy can be related to the surface tension. Therefore the Gibbs-Duhem equation is rearranged to... 

Gibbs [2] derived a thermodynamic relationship between the surface or interfacial tension y and the surface excess F (adsorption per unit area). The starting point of this equation is the Gibbs-Duhem equation, as given above [see Eq. (5.1)]. At equilibrium, where the rate of adsorption is equal to the rate of desorption, dG = 0. Hence, at a constant temperature, but in the presence of adsorption. 

Using the Gibbs model, it is possible to obtain a definition of the surface or interfacial tension y. The surface free energy dG comprises three components (i) an entropy term S dT (ii) an interfacial energy term Ady, and (iii) a composition term S d/t (where W is the number of moles of component i with chemical potential nf. The Gibbs-Duhem equation is,... 

In the framework of the membrane approach, the film can be treated as a single surface phase, whose Gibbs-Duhem equation reads "... 

Similar to the derivation of the Gibbs-Duhem equation, it is also possible to show the dependence of surface tension on the chemical potentials of the components in the interfacial region. If we integrate Equation (201) between zero and a finite value at constant A, T and nb to allow the internal energy, entropy and mole number to almost from zero to some finite value, this gives... 

Motomura et al. proposed a method of evaluation of various thermodynamic properties of micellar solutions from the surface tension data in the framework of the pseudophase treatment of micellisation [50, 52-55]. According to these authors, the micellar composition at the CMC can be found from the functional dependence of the CMC on the overall surfactant mole fraction using an analogy to the method proposed by Nguyen et al. [49], The approach of Motomura et al. [50] gives also a possibility to determine the relation between the composition of the surface layer and the micelles. Application of the Gibbs-Duhem equation to the whole... 

We must include all variables that contribute to similarity. For instance, if in a system the surface energy is relevant, the surface does not increase linearly with the volume of a similar system and a Gibbs - Duhem equation does not hold longer. 

The second quantity obtained from the Gibbs-Duhem equation is the relative surface excess which is the difference of the amount of a substance in the interphase and the relative amount of solvent (s) expressed by the ratio of the mole fractions r, and x. ... 

Surface solution theories should be consistent with the Gibbs adsorption equation (Equation 1.24), just as the bulk solution theories should be consistent with the Gibbs-Duhem equation. It is readily shown that ideal surface solution theory, as discussed above, satisfies this criterion (see Problem 1.15). However, regular surface solution theory does not, owing to the limitations of modeling the interfacial region by a single composition. This matter is discussed in some detail by Defay et al. (1966). 

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