Electrocapillary curves equation

Equation V-64 is that of a parabola, and electrocapillary curves are indeed approximately parabolic in shape. Because E ax tmd 7 max very nearly the same for certain electrolytes, such as sodium sulfate and sodium carbonate, it is generally assumed that specific adsorption effects are absent, and Emax is taken as a constant (-0.480 V) characteristic of the mercury-water interface. For most other electrolytes there is a shift in the maximum voltage, and is then taken to be Emax 0.480. Some values for the quantities are given in Table V-5 [113]. Much information of this type is due to Gouy [125], although additional results are to be found in most of the other references cited in this section. 

It follows that the surface excess F of an anion / (e.g. the Cl ion) can be evaluated from the electrocapillary curves of a given electrolyte (e.g. HCl) by plotting surface tension against the logarithm of the activity of the electrolyte (evaluated at various constant potentials) and determining the slope of the curve dy/d log and introducing it into equation 20.7. 

The excess charge on the electrode can be obtained from the slope of the electrocapillary curve (at any potential), by the Lippman equation ... 

A schematic example is given in Fig. A.4.5. The slope of the electrocapillary curve depends on the nature of the solution or the equilibrium structure of the double layer and on the specific sorbability of dissolved substances. In line with the Gibbs equation (Eq. 4.3), sorbable species depress the interfacial tension. 

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). 

Another interpretation of the electrocapillary curve is easily obtained from Equation (89). We wish to investigate the effect of changes in the concentration of the aqueous phase on the interfacial tension at constant applied potential. Several assumptions are made at this point to simplify the desired result. More comprehensive treatments of this subject may be consulted for additional details (e.g., Overbeek 1952). We assume that (a) the aqueous phase contains only 1 1 electrolyte, (b) the solution is sufficiently dilute to neglect activity coefficients, (c) the composition of the metallic phase (and therefore jt,Hg) is constant, (d) only the potential drop at the mercury-solution interface is affected by the composition of the solution, and (e) the Gibbs dividing surface can be located in such a way as to make the surface excess equal to zero for all uncharged components (T, = 0). With these assumptions, Equation (89) becomes... 

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. 

See also - electrocapillarity, - electrocapillary curve, -r Gibbs-Lippmann equation, - Wilhelmy plate (slide) method, - ring method, - Lippmann capillary electrometer. 

Ring method — Method to determine the - interfacial tension in liquid-gas systems introduced by Lecomte du Noiiy [i]. It is based on measuring the force to detach a ring or loop of a wire from the surface of a liquid. The method is similar to the -> Wilhelmyplate method when used in the detachment mode [ii]. See also -> electrocapillarity, -r electrocapillary curve, -> Gibbs-Lippmann equation, - Wilhelmy plate (slide) method, - drop weight method, - Lippmann capillary electrometer. 

Under the assumptions employed, the parabolic shape of electrocapillary curves in this model is expressed as the hyperbolic cosine of Aq Electrocapillary curves calculated as 25°C and z = 1 using Equation (15) are shown for ec/so = 8 and 80, where So is the vacuum permittivity, as dashed lines in Figures 7.2a and 7.2b, respectively. It is seen that the curvature of the electrocapillary curves becomes greater with increasing value of sc. 

Equations 49H and 50H explain why there has been little interest in obtaining the electrocapillary curve for ideally nonpolarizable interphases. On the other hand, this analysis can give us a feel for the type and magnitude of error that may arise when measurements are conducted with an electrode that is presumed to be ideally polarizable but in fact does allow some faradaic current to flow across the interphase. 

Lippmann equation (for electrocapillary curves) (1.5.6.17), 1.5.100, 1.5.108, 3.138 liquid junction potentials see potential difference liquid-liquid interface. 

Relationship (19.4), known as Lippmann equation, permits the evaluation of the excess of charge at the electrode surface from the electrocapillary curve y — y 4 )- Fot interfaces relatively simple such as the mercury/lM aqueous KCl interface, Eqn (19.4) results in a parabola with a maximum at dy/d0 = 0, i.e., for null charge at the electrode surface. This condition corresponds to the potential of zero charge (Pp ) for the electrode in the electrolyte solution. 

In agreement with the Lippmann equation, in the absence of surfactants the curve showing the surface tension as a function of the potential difference between phases (the electrocapillary curve) contains a maximum at some particular value of (p (Fig. Ill-17). This potential, which corresponds to the... 

In agreement with the Lippmann equation (III.21), the differentiation of the electrocapillary curve, o(cp), with respect to cp yields the surface charge density as a function of the surface potential, the second differentiation yields the value of the differential capacity, which can be compared with the results of the EDL theory. Based on such a comparison one can draw conclusions with respect to the validity of theoretical models and look for ways for further improvements. 

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. 

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... 

The principal interrelations of the basic quantities are well-known and explained in earlier reviews. " The general approach is based on 3D electrocapillary curve, with its two planar intersects corresponding to two Lippmann equations for constant chemical potentials pn = const and pn+ = const, see Refs. 41 3 for additional explanations. Here we mention the most important ratio ... 

The interfacial tension is a measurable quantity (see section 5 typical set of electrocapillary curves for a variety of electrolytes is Fig. 5.4. The curves are approximately parabolic and pass through a value called the electrocapillary maximum. Examination of the equation shows that the maximum corresponds to the condition electrode has no excess charge. At more negative potentials, the surface has a negative excess charge, and at more positive potentials positive surface charge. 

Oil/water interfaces are classified into the ideal-polarized interface and the nonpolarized interface. The interface between a nitrobenzene solution of tetrabutylam-monium tetraphenylborate and an aqueous solution of lithium chloride behaves as an ideal-polarized interface in a certain potential range. Electrocapillary curves of the interface were measured. The results are analyzed using the electrocapillary equation of the ideal-polarized interface and the Gouy-Chapman theory of diffuse double layers. The electric double layer structure consisting of the inner layer and the two diffuse double layers on each side of the interface is discussed. Electrocapillary curves of the nonpolarized oil/water interface are discussed for two cases of a nonpolarized nitrobenzene/water interface. 

These equations clearly show that the the slope of the electrocapillary curve of nonpolarized interface does not give the surface charge density but the relative surface excess of ionic components, as defined by Eq. (18) for case Ilb. In other words, the electrocapillary maximum potential does not correspond to the potential of zero charge . An approach to investigate the surface charge density and the double layer structure may be predicted as follows. When the values of the second terms of the right-hand sides of Eq. (18) (that is, the and Tnb values), are known or estimated on reasonable argument, Fd and F(so that by Eq. (19)) can be found from the slope... 

The charge defined by Equation 1.97 cannot be read directly as the slope of the electrocapillary curve, especially when the monolayer desorbs upon complex-ation. Indeed, due to the charge-transfer reaction taking place during the desorption of the complex, it is impossible to vary the potential E while keeping constant the chemical potential of the phospholipid L . Hence, the differentiation of the interfacial tension with respect to the potential does not give the surface charge... 

The effect of adsorption of surfactants at the mercury electrode on electrocapillary curves, drop times, maximum suppression have been discussed by Malik et al [227] and by Barradas and Kimmerle [228], the former attempting to measure the CMCs of non-ionic surfactants by changes in their effects on electrocapillary curves. By tagging micelles with a cation which is reducible at the dropping mercury electrode, Novodoff et al. [229] have developed a method of measuring the diffusion coefficient of the surfactant system using the Ilkovic equation,... 

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