Electrocapillary equation

Thermodynamic analysis of the ideally polarizable ITIES in the absence of the ion association yields the electrocapillary equation (for T,p = const) (Kakiuchi and Senda, 1983)... 

Every interface is more or less electrically charged, unless special care is exercised experimentally [26]. The energy of the system containing the interface hence depends on its electrical state. The thermodynamics of interfaces that explicitly takes account of the contribution of the phase-boundary potential is called the thermodynamics of electrocapillarity [27]. Thermodynamic treatments of the electrocapillary phenomena at the electrode solution interface have been generalized to the polarized as well as nonpolarized liquid liquid interface by Kakiuchi [28] and further by Markin and Volkov [29]. We summarize the essential idea of the electrocapillary equation, so far as it will be required in the following. The electrocapillary equation for a polarized liquid-liquid interface has the form... 

The difference s in the inner potentials is not directly measurable however, if the solution is in contact with a suitable reference electrode, its inner potential with respect to this electrode is fixed, and d 4>m — 4>s) — d, where (j) is the electrode potential. The resulting equation is known as the electrocapillary equation ... 

The electrocapillary equation (16.12) makes it possible to measure the surface excess of a species through ... 

Actually, in the electrocapillary equation for a solid electrode, the first and last terms are strictly given by... 

The election of a reference electrode sensitive to one of the ions of the electrolyte leads to the appearance in Eq. (1.71) of the surface excess of the other. Equation (1.71) is usually named as Lippman s electrocapillary equation. 

The fact that the Lippmann equation is the derivative of the electrocapillary equation shows that the charge aM is zero when the slope of the electrocapillary curve is zero. The potential where this occurs is called the point of zero charge, Ez, and occurs at the maximum in the electrocapillary curve, see Fig. 3.3. 

A further thermodynamic expression for l is possible.4 Since the electrocapillary equation (Eq. 8) is a total differential equation, the second cross-partial-differential coefficients of y are equal ... 

This equation can be used to determine V if the differential capacity C is extrapolated to zero frequency, namely under the quasi-equilibrium conditions required to apply the electrocapillary equation for a thermodynamic estimate of dE/dE (see Eq. 12) ... 

As the electrocapillary equation at constant pressure and temperature has the form... 

The surface tension of an electrode in contact with solution depends on the metal-solution potential difference. The equation describing this dependence is called the electrocapillary equation. It follows from the Gibbs adsorption isotherm, as we shall show in a moment. Before we do that, however, let us write this equation and discuss some of its consequences. 

Equation 6H is the electrocapillary equation in general form. To fully appreciate the possible applications of this equation and its limitations, we must derive it for a specific system. We have chosen a relatively simple case, to demonstrate the principles involved. Once these are understood, a more complex case can be derived following the same line of reasoning. 

It is interesting to see how the electrocapillary equation is modified if the interphase is ideally nonpolarizable. To do this, let us consider the cell... 

The third important relationship that follows from the electrocapillary equation is ... 

After this rather lengthy derivation, we should have arrived at the electrocapillary equation in its final fonn. Equation 23H does indeed look much like Eq. 6H, which we set out to derive. There is, however, another subtle point that we must deal with. One of the important uses of the electrocapillary equation, we recall, is to determine the surface excess of various species, in the present example that of ions. It would seem that Eq. 23H is just what we need, since it follows from this relationship that ... 

We shall end this section by using the electrocapillary equation to derive some relationships between partial derivatives, which are occasionally used in research in this field. Starting with a simplified form of Eq. 6H... 

For the ideally polarizable interphase, they are all independent. For the ideally nonpolarizable interphase, only two can be controlled independently. We recall that an ideally nonpolarizable electrode is a reversible electrode. By setting the concentrations (more accurately, the activities) of ions in the two phases, we determine the potential. Alternatively, by selling the potential, we determine the ratio of concentrations of this ion in the two phases. We conclude that the electrocapillary equation for the nonpolarizable interphase must have one less degree of freedom. 

We begin by developing the Gibbs adsorption isotherm, which describes interfaces in general, and from that we obtain the electrocapillary equation, which describes the properties of electrochemical interfaces more particularly. 

Other systems would have similar equations involving terms for other components. More general statements of the electrocapillary equation are available in the specialized literature (4). 

It is not obvious why (13.1.31) is called an electrocapillary equation. The name is a historic artifact derived from the early application of this equation to the interpretation of measurements of surface tension at mercury-electrolyte interfaces (1-4, 6-8). The earliest measurements of this sort were carried out by Lippmann, who invented a device called a capillary electrometer for the purpose (9). Its principle involves null balance. The downward pressure created by a mercury column is controlled so that the mercury-solution interface, which is confined to a capillary, does not move. In this balanced condition, the upward force exerted by the surface tension exactly equals the downward mechanical force. Because the method relies on null detection, it is capable of great precision. Elaborated approaches are still used. These instruments yield electrocapillary curves, which are simply plots of surface tension versus potential. 

Again, we take up the specific chemical system discussed in Section 13.1.2. From its electrocapillary equation, (13.1.31), it is clear that... 

Returning now to the electrocapillary equation, (13.1.31), we find that the relative surface excess of potassium ion at the interface considered there is given by (1, 6-8)... 

Here, coordinate z is normal to the interface, and the integration is performed over the whole perturbed region. An important point in this formula is the lower integration limit, ZG. corresponding to the choice of the phase boundary , whose attribution is not self-evident. This ambiguity of the definition (66) is removed by the thermodynamics of charged interfaces based on the Gibbs electrocapillary equation (67) ... 

If the concept of partial charge is introduced into the electrocapillary equation one gets the following result for the relative surface excess of the adsorbed ion ... 

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