Equation Grahame

Equation (69), known as the Grahame equation, describes the variation of charge density with potential at the surface with no limitations as to the value of the potential. Example 11.2 considers an application of this relationship. 

Figure 4.4 Surface potential-versus-surface charge calculated with the full Grahame equation (4.26, continuous line) and with the linearized version Eq. (4.27, dotted).

The integral can be solved with the help of Grahames equation (4.26) ... 

In order to estimate the surface charge we use the Grahame equation a = a/8 c eeff T-smh ertl)o/2kBT). With... 

Another approach to develop the Stern-Grahame equation is described by Fuerste-nau and Raghavan35) who apply the law of mass action to a linear mixture of similarly sized molecules showing ideal behavior in both the liquid phase and in the adsorption layer ... 

At the concentration (CHm) when hemi-micelles start to form it is possible to use the Stern-Graham equation for calculation of the energy

group from the water phase to the hemi-micelle102) ... 

The adsorption heat can be calculated using the Stern-Graham Equation, as shown by Ball and Fuerstenau157 ... 

A direct determination of the adsorption density of surfactants as dependent on Na+, Ca+, and Fe3+ concentrations was performed by Dobias179). His results are shown in Fig. 17. To interpret the results of adsorption measurement, he used the Stem-Graham equation (Eq. 51). Under the assumption that Na+, Cl-, Br- and cetyltrimethylam-monium ion (the surfactant used) are not the PDI, he modified the equation to get the form... 

The general methods of synthesis of diaziridines and diazirines were worked out by Schmitz (Equation (12)) <84CHEC-i(7)i95> and Graham (Equation (13)) <65JA4396> at the beginning of the study of these classes of compounds. 

The theory of EDL provides a connection between surface charge and surface potential (known as the Gouy equation or Graham equation ), which can be presented in the form ... 

The Gouy-Chapman portion of the model is expressed in the Muller equation, which is commonly but erroneously referred to as the Grahame equation (Tatulian, 1999),... 

Because of this, it is possible to assume that under conditions of low potentials the charge density and its potential at the base of the diffuse layer are tied by Grahame equation... 

Grahame equation and also as the contact theorem [6]. This, fundamentally, is a relationship between the surface charge density, (Tq (which is defined as o-q = — Jpedy, with a SI unit of C/m ), and the limiting value of the ionic density profile at the substrate-fluid interface. For a single fiat surface with an infinite extent of the adjacent liquid, an expression for co can be obtained from the Poisson-Boltzmann equation as... 

For the flat surfaces of a nanochannel, the surface charge density can be approximated using the Grahame equation. Especially, for the monovalent electrolyte solution, the surface charge density can be simply described by... 

The surface potential is related to the surface charge density through the charge balance, also known as the Grahame equation ... 

Once the zeta potential has been found, it can be used to calculate the surface charge density using the Grahame equation. 

The first direct observation of oxidative addition of the C-H bond of a saturated hydrocarbon to a transition metal center was reported in 1982 by Janowicz and Berman and by Hoyano and Graham (Equation 6.27). Jones reported the oxidative addition of alkane C-H bonds to the analogous rhodium complexes at nearly the same time (Equation 6.28). These reports have provided the foimdation for himdreds of subsequent reports of C-H bond cleavage by late transition metal complexes and studies of the mechanism by which a metal can cleave an alkane C-H bond under mild conditions. In fact, cleavage of the strong C-H bond in methane - (104 kcal/mol) occurs even at 12 K in a CH matrix. ... 

Why do alkyltrimethylammonium surfactants form roughly spherical aggregates on silica but cylindrical aggregates on mica The answer probably lies in the far greater density of electrostatic adsorption sites available on mica. Zeta potential measurements [40] in 1 mM KCl have shown that at pH 6 (a typical imaging condition), the surface potentials % of silica and mica are —80 and —120 mV, respectively. Using the Grahame equation... 

The electrostatic double-layer force can be calculated using the continuum theory, which is based on the theory of Gouy, Chapman, Debye, and Hiickel for an electrical double layer. The Debye length relates the surface charge density of a surface to the electrostatic surface potential /o via the Grahame equation, which for 1 1 electrolytes can be expressed as... 

The above equation is a general result, known as the Grahame equation, and can be used to assess the accumulation of various ions at a charged interface essentially as a monolayer. 

In nonpolar media, less charge is needed to reach a certain surface potential. Quantitatively, this is deduced from the Grahame equation (4.32). At low potentials, the surface potential is given by Eq. (4.33) ... 

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