Thermodynamics 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... 

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) ... 

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) ... 

A. Thermodynamics of the Electrocapillary Effect The basic equations of electrocapillarity are the Lippmann equation [110]... 

The Lippmann Equation. It can be shown thermodynamically that the slope of the electrocapillary curve is equal to the charge density, a, in the electric double layer (First Lippmann Equation). 

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]... 

It is impossible to write an advanced text in any area of physical chemistry without resort to some mathematical derivations, but these have been kept to a minimum consistent with clarity, and used mostly when several steps in the derivation involve approximations, or some other physical assumption, which may not be obvious to the reader. Thus, the theories of the diffuse-double-layer capacitance and of electrocapillary thermodynamics are derived in some detail, while the discussion of the diffusion equation is limited to the translation of the conditions of the experiment to the corresponding initial and boundary conditions and the presentation of the final results, while the sometimes tedious mathematical methods of solving the equations are left out. The mathematical skills needed to comprehend this book are minimal, and it should be easily followed by anybody with an undergraduate degree in science or engineering. An elementary knowledge of thermodynamics and of chemical kinetics is assumed, however. 

Several other derivations of (23) are possible. A fairly simple derivation, of a more thermodynamic nature, may be given in connection with the theory of the electrocapillary curve. In that case we must introduce a new set of variables. We start from the well known Lippmann equation... 

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