Gibbs adsorption equation, extended

Takahashi et al.67) prepared ionene-tetrahydrofuran-ionene (ITI) triblock copolymers and investigated their surface activities. Surface tension-concentration curves for salt-free aqueous solutions of ITI showed that the critical micelle concentration (CMC) decreased with increasing mole fraction of tetrahydrofuran units in the copolymer. This behavior is due to an increase in hydrophobicity. The adsorbance and the thickness of the adsorbed layer for various ITI at the air-water interface were measured by ellipsometry. The adsorbance was also estimated from the Gibbs adsorption equation extended to aqueous polyelectrolyte solutions. The measured and calculated adsorbances were of the same order of magnitude. The thickness of the adsorbed layer was almost equal to the contour length of the ionene blocks. The intramolecular electrostatic repulsion between charged groups in the ionene blocks is probably responsible for the full extension of the... 

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 ... 

The reason for the reduction in y to very low values when using two surfactants can be understood from a consideration of the Gibbs adsorption equation, which may be extended for multicomponent systems (11), as follows ... 

Interaction Potential Between Two Flat Plates Using the Extended Gibbs Adsorption Equation... 

An alternative and insightful way to obtain the interaction potential is from the extended Gibbs adsorption equation [6-8]. The natural thermodynamic potential to describe the system depicted in Fig. 2.2 is the grand potential Q(r, V, p, h)... 

Applying exactly the same line of reasoning as for the derivation of the extended Gibbs adsorption equation for two flat plates, see (2.9), we now obtain... 

The force method and the extended Gibbs adsorption equation can also be applied to obtain the depletion interaction between a sphere and a flat plate. For the Gibbs adsorption route we use (again)... 

The limitation of the Deijaguin approximation is that it only provides reliable results for R > Rg. To obtain results for the interaction potential between spheres for arbitrary q = Rg/R we use the extended Gibbs adsorption equation. Taniguchi et al. [33] and, independently, Eisenriegler et al. [34] found the concentration profile of Gaussian ideal polymer chains around a single hard sphere with radius R which reads... 

Application of the extended set of tools allows isotherms of binary systems to be computed, on the basis of Eq. (8) and the Gibbs adsorption equation, without any direct adsorption experiment. 

An alternative interpretation of the electrochemical double layer comes from a more thermodynamic approach. As an initial point, considering the Gibbs adsorption equation proved useful. This equation originally describes the dependence of the surface tension on the two-dimensional surface concentration (the surface excess F) of adsorbed particles as well as on their chemical potential p. The equation can be extended by introducing an electric term which considers the potential dependence of the surface tension. The Gibbs adsorption equation in its complete form is as follows ... 

Of these, (1) and (ii) have been encountered over and again. Adsorption isotherms were discussed in some detail in chapters 1.3 and II. 1 and 2. Appendix 1 of Volume II gives a survey of the most relevant isotherm equations. The corresponding 2D equations of state are repeated and extended in table 3.3 in sec. 3.4e. Now we consider set (lii). For each Isotherm is fully determined by and Tj(Cj), but as we also have the Gibbs equation, relating to changes in x and there is redundancy x(Cj) can be obtained in more than one way. 

Here and f are the extents of adsorption reactions which imply an independent transport of each constituent to the Gibbs geometrical surface from the bulk phases b and b such that dnf = d + d Gibbs surface system model is used and the surface phase considered also contains parts of the homogeneous phases. Therefore no direct comparison between Eqs. (32) and (42) is possible. The last-mentioned equation cannot, however, be in contradiction to our assumptions because the regarded surface phase also contains homogeneous masses it represents instead an alternative way of extending the theory to nonequilibrium systems. 

Generally, wastewaters are complex mixtures of solutes, which require theoretical approaches to predict multicomponent adsorption equilibria flxtm pure component adsorption data. The Ideal Adsorbed Solution model (IAS) was first established for a mixed gas adsorption by Myers and Prausnitz [9], and then extended to a multi-solute adsorption from dilute liquid solution by Radke and Prausnitz [10]. The model is based on the fundamental hypothesis that the multicomponent solution has the same spreading pressure s as that of the ideal single solution of the i component, the spreading pressure being the difference between the interfacial tension of the pure solvent and that of the solution containing the solute. This hypothesis is described by the Gibbs equation ... 

The requirement of constant composition poses a problem for solutions and other mixtures. When the interfacial area is extended, adsorption will occur at the newly formed interface. To relate the interfacial tension to the excess interfacial Gibbs (or, for that matter, Helmholtz) energy, adsorbed amounts and concomitant changes in the composition of the bulk phases have to be established, as has been discussed in Section 3.9 (Equation 3.77). Further elaboration on the relation between interfacial tension, adsorbed amounts, and composition of the bulk phase can be found in Chapters 7,11, and 14. 

The most straightforward (and the most developed) approach to multicomponent adsorption is in further development of the thermodynamics of a surface phase, similar to the bulk-phase thermodynamics. In this way, the Gibbs surface thermodynamics should be completed by an equation of state or by an excess model for a proper thermodynamie potential. An extended review of the fundamentals and the history of the development of this approaeh may be found in Refs. 8, 9, and 78. The approach has become espeeially popular and widely used for praetieal modeling of multicomponent adsorption after the works of de Boer [79] and, especially, Myers and Prausnitz [80]. The latter authors made the natural step of introducing the activity coefficients y of the components in an adsorbed phase. In terms of these coefficients, the chemical potentials of the adsorbate may be expressed as... 

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