Gibbs-Duhem equation interface

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 ... 

As a first example, consider a pure liquid in equilibrium with its vapor. Because I wish to focus attention on the liquid/gas interface to the exclusion of adsorption effects at solid boundaries, I shall suppose the containing vessel to be chemically inert. The Gibbs-Duhem equation for the system is then... 

The latter fact may be emphasized by considering small samples of molar content n and n3 drawn from the bulk phases in regions far from the interface. The size and shape of these samples need have no relationship to the geometry of the interface—any irregularly shaped specimen of bulk phase will do. For each of these bulk phase samples a and p we have Gibbs-Duhem equations... 

The equilibrium bubble size can be determined based on applying the pressure balance at the bubble interface. Gibbs-Duhem equation and Young-Laplace equation are written for the bubble [1],... 

Formnately such problems can usually be avoided — at least for plane interfaces. The Gibbs-Duhem equations for the two bulk phases in a binary system at constant temperature are given by... 

Since interfaces are binary or multi-component systems, their thermodynamics can be described by a Gibbs-Duhem-equation ... 

The thermodynamic theory of the ideally polarised electrode has been extensively reviewed in the past few decades [1-5], and the relationship with the ideally non-polarisable interface has been derived in an elegant treatment by Parsons [6]. The starting point in all derivations is the Gibbs-Duhem equation which defines the relationship between the extensive thermodynamic variables. For a bulk phase this has the form ... 

Example 2.8 Derive the following Gibbs-Duhem equation for an open phase with a curved interface ... 

Note that the Gibbs-Duhem equation for a bulk phase does not depend on the interface it is the same whether the interface is flat or curved. In other words, the Gibbs-Duhem equation of the bulk phase, 6, is always... 

In this equation and F are real excesses in the interface. At issue now is whether these are identical to the analytically determined ones, using some depletion technique. The problem is that in such methods one measures the decrease of the solute concentration by adsorption, but how can one do that in practice when the system contains more solute them solvent We discussed this issue in detail in secs. II.2.1 and 4. Let us briefly elaborate this for consfaint temperature. In [4.2.1] d/Xj and d/x are coupled by the Gibbs-Duhem relation (l-x)d/Xj -i-xd/x =0, so we can eliminate either one of the chemieal potentials, obtaining these alternatives... 

Solution of the PB equation with these boundary condition enables one to evaluate the spatial distribution of electric potential, field, and ion concentration. Knowing these quantities, one can evaluate the force and energy of interactions between colloid particles or between particles and interfaces. This can be achieved via the thermodynamic equilibrium condition (Gibbs-Duhem relationship) which can be formulated as [13]... 

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