Volmer equation

We have addressed the various adsorption isotherm equations derived from the Gibbs fundamental equation. Those equations (Volmer, Fowler-Guggenheim and Hill de Boer) are for monolayer coverage situation. The Gibbs equation, however, can be used to derive equations which are applicable in multilayer adsorption as well. Here we show such application to derive the Harkins-Jura equation for multilayer adsorption. Analogous to monolayer films on liquids, Harkins and Jura (1943) proposed the following equation of state ... 

Various other non-ideal-gas-type two-dimensional equations of state have been proposed, generally by analogy with gases. Volmer and Mahnert [128,... 

Equations such as V-96 are known as Butler-Volmer equations [150]. At equilibrium, there will be equal and opposite currents in both directions, =... 

Equation XVII-70 bears a strong resemblance to the Langmuir equation (see Ref. 4)—to the point that it is doubtful whether the two could always be distinguished experimentally. An equivalent form obtained by Volmer [53] worked well for data on the adsorption of various organic vapors on mercury [54] (see Problem XVII-40). 

As tire reaction leading to tire complex involves electron transfer it is clear that tire activation energy AG" for complex fonnation can be lowered or raised by an applied potential (A). Of course, botlr tire forward (oxidation) and well as tire reverse (reduction) reaction are influenced by A4>. If one expresses tire reaction rate as a current flow (/ ), tire above equation C2.8.11 can be expressed in tenns of tire Butler-Volmer equation (for a more detailed... 

Buthiazide [2043-38-1] Buthidazole [55511-98-6] Buthiobate [51308-54-4] Buthiomne sulfoximme Butisol sodium Buder-Volmer equation... 

The Langmuir model is discussed in reference 19 the Volmer in reference 20 and the van der Waals and virial equations in reference 8. 

In order to clear up the mechanism of inactivation of excited states, we examined the processes of quenching of fluorescence and phosphorescence in PCSs by the additives of the donor and acceptor type253,2S5,2S6 Within the concentration range of 1 x 1CT4 — 1 x 10"3 mol/1, a linear relationship between the efficiency of fluorescence quenching [(/0//) — 1] and the quencher concentration was found. For the determination of quenching constants, the Stem-Volmer equation was used, viz. 

The exchange current density for common redox couples (at room temperature) can range from 10-6 pAcm-2 to A cm"2. Equation (1-24) can be written in terms of the exchange current to give the Butler-Volmer equation ... 

Background current, 21, 65 Background subtraction, 40, 106 Bacteria electrode, 182 Band microelectrodes, 130, 135 Beryllium, 82 Bienzyme electrodes, 175 Biocatalytic devices, 172 Biological recognition, 171 Biosensors, 50, 171 Bipotentiostat, 106 Blood electrolyte, 165 Boltzmann equation, 19 Brain analysis, 40, 116 Butler-Volmer equation, 14... 

This is the relaxation time of the polymer oxidation under electro-chemically stimulated conformational relaxation control. So features concerning both electrochemistry and polymer science are integrated in a single equation defining a temporal magnitude for electrochemical oxidation as a function of the energetic terms acting on this oxidation. A theoretical development similar to the one performed for the Butler-Volmer equation yields... 

The activation overpotential Tiac,w is due to slow charge transfer reactions at the electrode-electrolyte interface and is related to current via the Butler-Volmer equation (4.7). A slow chemical reaction (e.g. adsorption, desorption, spillover) preceding or following the charge-transfer step can also contribute to the development of activation overpotential. 

Thermodynamic and mechanical equilibrium on a curved vapor-liquid interface requires a certain degree of superheat in order to maintain a given curvature. Characteristics of homogeneous and heterogeneous nucleation can be estimated in the frame of classical theory of kinetics of nucleation (Volmer and Weber 1926 Earkas 1927 Becker and Doring 1935 Zel dovich 1943). The vapor temperature in the bubble Ts.b can be computed from equations (Bankoff and Flaute 1957 Cole 1974 Blander and Katz 1975 Li and Cheng 2004) for homogeneous nucleation in superheated liquids... 

I. Development of a simple, Butler-Volmer equation-based kinetic model for MiXi (CdTe) electrodeposition. J Electrochem Soc 132 2904-2909... 

Figure5.9 Spectral changes upon addition ofCH3OH to aCH2Cl2 solution of [(8-QNS)2Au(AuPPh3)2]BF4 (7.72 x 10 M) with excitation at 320 nm. Inset Changes in lifetime according to Stern-Volmer equation. Reproduced with permission from [35]. Copyright (2003) American Chemical Society.

In 1930, Max Volmer and Tibor Erdey-Griiz used the concept of a slow discharge step for cathodic hydrogen evolntion (slow discharge theory). According to these ideas, the potential dependence of electrochemical reaction rate constants is described by Eq. (6.5). Since hydrogen ions are involved in the slow step A, the reaction rate will be proportional to their concentration. Thus, the overall kinetic equation can be written as... 

Experimental studies of electrode kinetics resulted in the formnlation of the basic empirical relationship, the Volmer-Butler equation, (6.10) or (6.13), describing the dependence of the electric current on the electrode potential. This eqnation involves the potential E, the rate constants, and the concentrations. 

Important parameters involved in the Volmer-Butler equation are the transfer coefficients a and (1. They are closely related to the Bronsted relation [Eq. (14.5)] and can be rationalized in terms of the slopes of the potential energy surfaces [Eq. (14.9)]. Due to the latter, the transfer coefficients a and P are also called symmetry factors since they are related to the symmetry of the transitional configuration with respect to the initial and final configurations. 

This method is applicable when the fluorescence of a ligand is quenched in presence of DNA or RNA and provides base-dependent specificity [135]. In fluorescence quenching experiments the titration data is plotted according to the Stern-Volmer equation ... 

No steady-state theory for kinetically controlled heterogeneous IT has been developed for micropipettes. However, for a thin-wall pipette (e.g., RG < 2) the micro-ITIES is essentially uniformly accessible. When CT occurs via a one-step first-order heterogeneous reaction governed by Butler-Volmer equation, the steady-state voltammetric response can be calculated as [8a]... 

The effect of the phospholipids on the rate of ion transfer has been controversial over the last years. While the early studies found a retardation effect [6-8], more recent ones reported that the rate of ion transfer is either not retarded [9,10] or even enhanced due to the presence of the monolayer [11 14]. Furthermore, the theoretical efforts to explain this effect were unsatisfactory. The retardation observed in the early studies was explained in terms of the blocking of the interfacial area by the phospholipids, and therefore was related to the size of the transferring ion and the state of the monolayer [8,15]. The enhancement observed in the following years was attributed to electrical double layer effects, but a Frumkin-type correction to the Butler Volmer (BV) equation was found unsuitable to explain the observations [11,16]. Recently, Manzanares et al. showed that the enhancement can be described by an electrical double layer correction provided that an accurate picture of the electrical double layer structure is used [17]. This theoretical approach will be the subject of Section III.C. 

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