Lippmann equation

See also -> adhesion, -> Dupre equation, -> Lippmann equation, -> Lippmann capillary electrometer, -> point of zero charge, -> Young equation. 

See also capillarity, adhesion, double layer, Dupre equation, Gibbs-Lippmann equation, electrocapillary equation, -> electrokinetic effects, Lipp-mann equation, Lippmann capillary electrometer, point of zero charge, Young equation, Wilhelmy... 

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

Kouri D J, Huang Y, Zhu W and Hoffman D K 1994 Variational principles for the time-independent wave-packet-Schrddinger and wave-packet-Lippmann-Schwinger equations J. Chem. Phys. 100... 

The basic equations to be used are the Lippmann-Schwinger equations for the alloy wave function... 

At constant p and T, the Gibbs adsorption equation for an electrode interface leads to the well-known Lippmann equation12 ... 

This equation was hrst obtained by Gabriel Lippmann in 1875. The Lippmann equation is of basic importance for electrochemistry. It shows that surface charge can be calculated thermodynamically from data obtained when measuring ESE. The values of ESE can be measured with high accuracy on liquid metals [e.g., on mercury (tf= -39°C)] and on certain alloys of mercury, gallium, and other metals that are liquid at room temperature. 

In 1873, Gabriel Lippmann (1845-1921 Nobel prize, 1908) performed extensive experiments of the electrocapiUary behavior of mercury and established his equation describing the potential dependence of the surface tension of mercury in solutions. In 1853, H. Helmholtz, analyzing electrokinetic phenomena, introduced the notion of a capacitor-like electric double layer on the surface of electrodes. These publications... 

For solutions of simple electrolytes, the surface excess of ions can be determined by measuring the interfacial tension. Consider the valence-symmetrical electrolyte BA (z+ = —z = z). The Gibbs-Lippmann equation then has the form... 

In this manner, the surface excess of ions can be found from the experimental values of the interfacial tension determined for a number of electrolyte concentrations. These measurements require high precision and are often experimentally difficult. Thus, it is preferable to determine the surface excess from the dependence of the differential capacity on the concentration. By differentiating Eq. (4.2.30) with respect to EA and using Eqs (4.2.24) and (4.2.25) in turn we obtain the Gibbs-Lippmann equation... 

The fact that the electrocapillary quantities are measured relative to their value in the base electrolyte can also be expressed in the formulation of the Gibbs-Lippmann equation. If quantities referred to the base electrolyte are primed and quantities referred to the studied surface-active substance are denoted by the subscript 1, then... 

The appearance of peaks on the differential capacity curves can be derived from this potential dependence in the following manner. The Gibbs-Lippmann equation (see Eq. 4.2.23) gives... 

G. Lippmann introduced the capillary electrometer to measure the surface tension of mercury (Fig. 4.10). A slightly conical capillary filled with mercury under pressure from a mercury column (or from a pressurized gas) is immersed in a vessel containing the test solution. The weight of the mercury column of height h is compensated by the surface tension according to the Laplace equation... 

A first step in attempting to explain the shape of the plot in Figure 2.1(d) is provided by the Lippmann equation ... 

If the interfacial tension 7 can be measured, the surface charge density can be obtained by differentiation, which yields the Lippmann equation ... 

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... 

General Time-to-Energy Transform of Wavepackets. Time-Independent Wavepacket-Schroedinger and Wavepacket-Lippmann-Schwinger Equations. 

Energy-Separable Polynomial Representation of the Time-Independent Full Green Operator with Application to Time-Independent Wavepacket Forms of Schrodinger and Lippmann-Schwinger Equations. 

Variational Principles for the Time-Independent Wave-Packet-Schrodinger and Wave-Packet-Lippmann-Schwinger Equations. 

Electrocapiilary phenomena on Hg-electrode in presence and absence of an adsorbate (camphor). From a measurement of interfacial tension (a) (e.g., from droptime of a Hg-electrode) or of differential capacity (d) (e.g., by an a.c-method) as a function of the electrode potential (established by applying a fixed potential across tine Hg-electrolyte interface) one can calculate the extent of adsorption (b) (from (a) by the Gibbs Equation) and of the structure of the interface as a function of the surface potential. Figs, a, c and d are interconnected through the Lippmann Equations. 

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). 

At the electrocapillary maximum, the charge density, a, is zero (point of zero charge) (Fig. A.4.5c). By definition, the differential capacity of the double layer, Cd, is equal (Second Lippmann Equation). 

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... 

Schofield Phil. Mag. March, 1926) has recently verified this relation by direct experiment. In order to appreciate the significance of this result, it is necessary to consider in more detail the electrical potential difference V and the manner in which it arises. Instead of regarding the phenomenon from the point of view of the Gibbs equation, it has been, until recently, more usual to discuss the subject of electro-capillarity from the conceptions developed by Helmholtz and Lippmann. These views, together with the theory of electrolytic solution pressure advanced by Nemst, are not in reality incompatible with the principles of adsorption at interfaces as laid down by Gibbs. 

Although there exists a concentration gradient of mercurous sulphate from the anode to the cathode it is unlikely that this causes an appreciable alteration in the electrical potential throughout the bulk of the electrolyte. Hence without serious error the applied E.M.F. V may be equated to the difference in the values of V at the two electrodes and since F at the unpolarisable electrode remains unaltered the variation of a with V is observed at the cathode. With the aid of thermo-dynamic reasoning, Lippmann deduced the relation... 

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