Lippmann relation

W. Kohlrausch, Leitfaden der Praktischen Physik, 31 ed BiblioBazaar, 1875. G. Lippmann, Relations entre les phenomenes electrique et capillaries, Atm. Chim. Phys., vol. 5, pp. 494-549,1875. 

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 ... 

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... 

From the Lippmann-Schwinger equation and the expansion of Go in some specified cell r/x, the coefficients in these expansions must be related by = cl + ft N jyf. Consistency conditions are derived by considering the alternative... 

These two Green functions are related by the Lippmann-Schwinger integral equation... 

The decomposition (A.20) is a particular case of the relations obtained in the problem of particle scattering by a potential (of the Lippmann-Schwinger type). It turns out to be particularly useful for the calculation of the states of impurities, or of impurity aggregates, in a perfect lattice whose propagator G0 is known. 

Conversion of values of y into capacities is done by double differentiation in relation to the electrostatic potential difference A0 between its value in the metal, 0M, and that in the solution, (j)s. The first derivative gives the charge on the interface, and is the Lippmann equation... 

Lippmann was the first to record an accurate study of the relation between electrical potential and surface tension, in the classical memoir1 in which the capillary electrometer is described. 

The relation between the change of tension, at constant applied potential, and the change of potential at constant interfacial tension, for a given change in concentration of any component, in either phase, can be obtained at once from Lippmann s equation, since... 

Electrocapillarity — (a) as a branch of science, this term covers all phenomena related to the thermodynamics of charged - interfaces, esp. of metal-solution interfaces. The term is practically synonymous with -> capillarity, but emphasizes the electric aspects, (b) The term electrocapillarity is often used in a restricted sense to mean the study of the equilibrium properties of metal solution interfaces, such as the - interfacial tension of mercury solution interfaces, the height of a mercury column (in the case of the - Lippmann capillary electrometer), or the -> drop time (in the case of the - dropping mercury electrode). More generally, however, the equilibrium properties of many other interfaces fall... 

The reduced Lippmann—Schwinger equations are obtained by expanding all the amplitudes of (6.73) according to (7.36) and again using the orthonormality relations (3.71,3.89) to eliminate the integral over k and the sum over Clebsch—Gordan coefficients in the expansion of the projection operator... 

Pugsley TA, Lippmann W. 1979. Effect of acute and chronic treatment of tandamine, a new heterocyclic antidepressant, on biogenic amine metabolism and related activities. Naunyn Schmiedebergs Arch. Pharmacol. 308 239 17... 

Lipfhrt, E W. 1989. Air pollution and materials damage. In The handbook of environmental chemistry, ed O. Hutzinger, vol. 4B, pp. 114-86, Berlin Springer-Verlag. Lippmann, E 1979. Stabilitatsbeziehungen der Tonminerale (Stability relations of clay minerals). N. Jb. Miner. Abh. l36(3) 287-309. 

In the specific case of a "strictly congruent" dissolution process occurring in an aqueous phase with a [B+]/[C+] activity ratio equal to the B+/C+ ratio in the solid, primary-saturation can be approximately found by drawing a straight vertical line on the Lippmann diagram from the solid-phase composition to the solutus (see figure 1). For an exact calculation, the following relations may be used to determine the primary saturation state ... 

Stoichiometric saturation states can be represented on Lippmann phase diagrams (figure 1) by relating the total solubility product variable Ellgg (defined specifically at stoichiometric saturation with respect to a solid Ex. CxA) to the Kgg constant (equation 12) and to the aqueous activity fractions and c,aq-... 

For liquid metals (mercury, gallium) or their alloys, one can measure another interfacial quantity, interfacial tension equal to the specific energy of the interface formation, y, at different values of the electrode potential, E[, 17]. At equilibrium the Lippmann equation relates it to the electrode-charge density a, that is, to the charge Q in Eq. (2) per unit surface area ... 

Lippmann G (1875) Relations entre les phenomenes electriques et capillaries. Ann Chim Phys 5 494—549... 

Size consistency of the Brillouin-Wigner perturbation theory is studied using the Lippmann-Schwinger equation and an exponential ansatz for the wave function. Relation of this theory to the coupled cluster method is studied and a comparison through the effective Hamiltonian method is also provided. 

Lu 3-010 (3,3-dimethyl-l-(3-methylaminopropyl)-l-phenylphthalan), a specific blocker of catecholamine uptake devoid of anticholinergic activity, inhibited both basal and pentagastrin stimulated secretion in the rat.88 studies by Lippmann on Lu 3-010 and another series of com-pounds 9 related to N,N -bis-(l-naphthylmethyl)-l, t-cyclohexane-bis-(methylamine) dihydrochloride (AY 9928) suggest that direct correlation between blockade of catecholamine uptake and antisecretoiy effect is moot. 

If an electric field is applied across the planar confinement, additional COTitributions reducing the wetting surface free energy (3) arise. Electrocapillarity can reverse the sign of cosB, leading to electrowetting of a lyophobic surface. Ignoring any field dependence of liquid/vapor (7) and solid/liquid (ysi) surface tensions, within continuum approximations, the macroscopic relation due to Lippmann [55] describes electrocapillarity by... 

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