Thermodynamics of ideal polarizable interfaces

For liquid electrodes thermodynamics offers a precise way to determine the surface charge and the surface excesses of a species. This is one of the reasons why much of the early work in electrochemistry was performed on liquid electrodes, particularly on mercury - another reason is that it is easier to generate clean liquid surfaces than clean solid surfaces. With some caveats and modifications, thermodynamic relations can also be applied to solid surfaces. We will first consider the interface between a liquid electrode and an electrolyte solution, and turn to solid electrodes later. 

Consider a single bulk phase with both charged and uncharged particles in equilibrium the differential dU of the internal energy is  

The simplest way to treat an interface is to consider it as a phase with a very small but finite thickness in contact with two homogeneous phases (see Fig. 16.1). The thickness must be so large that it comprises the region where the concentrations of the species differ from their bulk values. It turns out that it does not matter, if a somewhat larger thickness is chosen. For simplicity we assume that the surfaces of the interface are flat. Equation (16.1) is for a bulk phase and does not contain the contribution of the surfaces to the internal energy. To apply it to an interface we must add an extra term. In the case of a liquid-liquid interface (such as that between mercury and an aqueous solution), this is given by 7 cL4, where 7 is the interfacial tension - an easily measurable quantity - and A the surface area. The fundamental equation (16.1) then takes on the form  

The quantities appearing in Eq. (16.2) are not independent. They are related by a Gibbs-Duhem equation, which is obtained in the same way as in the ordinary thermodynamics of bulk phases integrating with respect to the extensive variables results in Ua —TSa — pVa + 7Aa + E/if Nf. Differentiating and comparing with Eq. (16.2) gives  

The interface is in contact with two bulk phases, the metal electrode (index m ) and the solution (index s). Formally, we consider the metal to be composed of metal atoms M, metal ions Mz+, and electrons e these particles are present both in the electrode and the interface, but not in the solution. On the other hand, certain cations and anions and neutral species occur both in the solution and the interface. Since the electrode is ideally polarizable, no charged species can pass through the interface. 

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A typical example of an ideal polarizable interface is the mercury-solution interface [1,2]. From an experimental point of view it is characterized by its electrocapillary curve describing the variation of the interfacial tension 7 with the potential drop across the interface, 0. Using the thermodynamic relation due to Lippmann, we get the charge of the wall a (-a is the charge on the solution side) from the derivative of the electrocapillary curve ... 

The situation that no charge transfer across the interface occurs is named the ideal polarized or blocked interface. Such interfaces do not permit, due to thermodynamic or kinetic reasons, either electron or ion transfer. They possess Galvani potentials fixed by the electrolyte and charge. Of course, the ideal polarizable interface is practically a limiting case of the interfaces with charge transfer, because any interface is always permeable to ions to some extent. Therefore, only an approximation of the ideal polarizable interface can be realized experimentally (Section III.D). 

The liquid metal mercury-solution interface presents the advantage that it approaches closest to an ideal polarizable interface and, therefore, it adopts the potential difference applied between it and a non-polarizable interface. For this reason, the mercury-solution interface has been extensively selected to carry out measurements of the surface tension dependence on the applied potential. In the case of other metal-solution interfaces, the thermodynamic study is much more complex since the changes in the interfacial area are determined by the increase of the number of surface atoms (plastic deformation) or by the increase of the interatomic lattice spacing (elastic deformation) [2, 4]. 

The thermodynamic properties of an ideally polarizable interface are most easily examined by considering an electrochemical cell with one polarizable electrode and one non-polarizable electrode. An example of such a system is... 

For many systems, the maximum of the electrocapillary curve is located in the region of ideal polarizability of the electrode, and the maximum potential corresponds to the potential of zero total (thermodynamic) and zero free charge. At the mercury-water interface, the region of ideal polarization is a few volts when soluble mercury salts are absent (2.2 V), but at the interface between two immiscible... 

In addition, the potential of the electrode can be varied, resulting in a change in the stmcture of the interface. If no current is passed when the potential of the electrode changes, the electrode is called an ideally polarizable electrode, and can be described using thermodynamics. 

The general thermodynamic approach yields the - Gibbs-Lippmann equation (- electrocapillary) for the nonpolarizable [v] and ideally polarizable [ix] ITIES. For the interface between the electrolyte solutions of RX in w and SY in o, see also - interface between two immiscible electrolyte solutions, this equation has the form [x]... 

Although the thermodynamic definition of an ideally polarized interface is unequivocal [28], the polarizability of an actual interface is understood differently, depending on what is to be measured at the interface. Gavach et al. [3] obtained the polarized range of c. 150 mV at the interface between an aqueous 10 M KCl solution and a nitrobenzene solution of dodecyltrimethylammonium dode-... 

The formation of 2D Meads phases on a foreign substrate, S, in the underpotential range can be well described considering the substrate-electrolyte interface as an ideally polarizable electrode as shown in Section 8.2. In this case, only sorption processes of electrolyte constituents, but no Faradaic redox reactions or Me-S alloy formation processes are allowed to occur, The electrochemical double layer at the interface can be thermodynamically considered as a separate interphase [3.54, 3.212, 3.213]. This interphase comprises regions of the substrate and of the electrolyte with gradients of intensive system parameters such as chemical potentials of ions and electrons, electric potentials, etc., and contains all adsorbates and all surface energy. Furthermore, it is assumed that the chemical potential //Meads is a definite function of the Meads surface concentration, F, and the electrode potential, E, at constant temperature and pressure Meads (7", ). Such a model system can only be... 

The thermodynamics of 2D Meads overlayers on ideally polarizable foreign substrates can be relatively simply described following the interphase concept proposed by Guggenheim [3.212, 3.213] and later applied on Me UPD systems by Schmidt [3.54] as shown in Section 8.2. A phase scheme of the electrode-electrolyte interface is given in Fig. 8.1. Thermodynamically, the chemical potential of Meads is given by eq. (8.14) as a result of a formal equilibrium between Meads and its ionized form Me in the interphase (IP). The interphase equilibrium is quantitatively described by the Gibbs adsorption isotherm, eq. (8.18). In the presence of an excess of supporting electrolyte KX, i.e., c , the chemical potential is constant and... 

Within the voltage limits set by the thermodynamic stability range of the electrolyte, foreign metal electrodes may sometimes be regarded as ideally polarizable or blocking. The metal electrodes must not react with the electrolyte, and for the moment adsorption and underpotential deposition will be neglected. From an electrochemical point of view, this is the simplest type of interface and has furnished much of the information we have about the electrified interface. 

The ideal non-polarized electrode is one which allows free and unimpeded exchange of electrons or ions across the electrode-solution interface. The reversible electrodes considered in an earlier chapter approximate in behaviour to these, the rapid establishment of thermodynamic equilibrium being consequent upon such rapid interchange. When charge crosses an ideally reversible electrode, the electrochemical changes which take place do so with such rapidity that the equilibrium situation is instantaneously restored. Such an electrode is non-polarizable in the sense that its potential, for small currents, has remained stable. This potential, for a fixed temperature and pressure, is... 

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