Lippmann equation solids

For solid electrodes the surface tension 7 must be replaced by the work done in forming a unit area of the metal by cleaving, which we also denote by 7. When a solid surface is charged, or when it is covered by adsorbates, surface strains can arise, and additional terms appear in the Lippmann equation. However, there is good reason to suppose that these terms are small [1], and the Lippmann equation is usually... 

A comparison with the reversible interface can be made. The reversible solid electrolyte interface can be used in a similar way to explore the distribution of charge components at solid-water interfaces. As we have seen, the surface charge density, o, (Eqs. (3.1) and (iii) in Example 2.1) can be readily determined experimentally (e.g., from an alkalimetric titration curve). The Lippmann equations can be used as with the polarized electrodes to obtain the differential capacity from... 

Surface stress — The surface area A of a solid electrode can be varied in two ways In a plastic deformation, such as cleavage, the number of surface atoms is changed, while in an elastic deformation, such as stretching, the number of surface atoms is constant. Therefore, the differential dUs of the internal surface energy, at constant entropy and composition, is given by dUs = ydAp + A m g m denm, where y is the interfacial tension, dAp is the change in area due to a plastic deformation, gnm is the surface stress, and enm the surface strain caused by an elastic deformation. Surface stress and strain are tensors, and the indices denote the directions of space. From this follows the generalized Lippmann equation for a solid electrode ... 

At the surface of a solid electrode the molecules are not free to move and the interface is in general not in thermodynamic equilibrium with the bulk. In addition, any real solid electrode is at least partially polarizable. These two effects lead to an additional superficial charge <7e. and the Lippmann equation becomes [26]... 

The additional charge and the corresponding additional surface tension are time-dependent quantities in which the equilibrium between the bulk and the interface is not established. The irreversible contribution can be separated from the reversible by considering the time dependence, if the experimental time scale allows for such a test. Time-dependent effects can be observed by impedance measurements at different frequencies. For gold, as an example, impedance measurements showed spectra characteristic for equilibration processes at least over a time scale of 0.1 ms to 100 s. Gold also shows a surface reconstruction depending on the potential [148]. Fortunately, the variation of the interfacial strain with potential is usually so small that the original Lippmann equation (41) for a solid is practically the same as for a liquid electrode 1149]. 

Electrowetting is the phenomenon of enhancing the wettability of a solid surface by applying an electric field, because the external electric field can alter the surface tension at the solid-liquid interface. This is expressed by the Lippmann equation [50] ... 

The equation used to calculate the value of Eli at primary saturation as a function of solid composition, for a strictly congruent dissolution process, may be found by combining the Lippmann solutus equation (4) with equations 7 and 8 ... 

Stoichiometric saturation states can be represented on Lippmann phase diagrams (figure 1) by relating the total solubility product variable Ellgg (defined specifically at stoichiometric saturation with respect to a solid Ex. CxA) to the Kgg constant (equation 12) and to the aqueous activity fractions and c,aq-... 

These equations define the families of ( ba ba) and ( cA> ncA) conditions for which a solution containing A-, B+ and C ions will be saturated with respect to pure BA and pure CA solids. Thermodynamic equilibrium with respect to a mechanical mixture of the two pure BA and CA solids, in contrast to a solid-solution of BA and CA, will be represented on a Lippmann diagram by a single point, namely the intersection of the pure BA and pure CA saturation curves. The coordinates of this intersection are ... 

The first verification of the fundamental equations of Gibbs and Lippmann was based on surface tension (surface energy) measurements. The investigation of solid electrodes with this method is restricted. In the following paragraph a description is given of some of the modem methods, which are used for investigation of the electrochemical double layer. 

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