Interfacial Gibbs-Duhem equation

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 ... 

The change of the interfacial tension can be calculated with the help of the Gibbs-Duhem equation even when a potential is applied. In order to use the equation, we first need to find out which molecular species are present. Evidently, only those which are free to move are of interest. In the electrolyte we have the dissolved ions. In the metal the electrons can move and have to be considered. 

By using the Gibbs model, it is possible to obtain a definition of the siuface or interfacial tension y, starting from the Gibbs-Duhem equation [2], that is... 

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,... 

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... 

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). 

Kammer [209] examined the interfacial phenomena of polymer melts from a thermodynamic point of view. A system of thermodynamic equations has been derived to describe the temperature, pressure, and composition dependence of interfacial structure. Starting from the fundamental equations of Guggenheim [210], Kammer employed the Gibbs-Duhem equation of intensive parameters (13) to find that the interfacial composition is given by ... 

The next step is to make clear the Gibbs adsorption amount, using the above relative adsorption. Recall (8.27), in which the interfacial tension is a function of i -H 2 independent variables. However, the Gibbs phase rule permits only i independent variables for two phases including i components. Therefore, the problem is how to reduce the number of intensive variables by two while keeping thermodynamical consistency. The Gibbs-Duhem equations for two homogeneous phases a and respectively, are... 

They define interfacial energy starting from the equation of Gibbs Duhem, which is then written for any transformation, as... 

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