Stern-Graham equation

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 ... 

Fuerstenau") was the first who used the Stern-Grahame model of EDL to describe the adsorption of long-chain surfactants for the equilibrium in heterogeneous systems. The adsorption density in the Stem plane is given by the equation... 

Equation 3.74 is a parameterization of ctm =/ (cp ) based on the Stern-Grahame double layer model. This relation requires as input the potential of zero charge and the Helmholtz capacitance. As mentioned, finding =f(

various surface and bulk oxide species, occurring simultaneously with double layer charging. 

Instead of an exact calculation, Gouy and Chapman have assumed that (4) can be approximated by combining the Poisson equation with a Boltzmann factor which contains the mean electrical potential existing in the interface. (This approximation will be rederived below). From this approach the distribution of the potential across the interface can be calculated as the function of a and from (2) we get a differential capacitance Cqc- It has been shown by Grahame that Cqc fits very well the measurements in the case of low ionic concentrations [11]. For higher concentrations another capacitance in series, Q, had to be introduced. It is called the inner layer capacitance and it was first considered by Stern [1,2]. Then the experimental capacitance Cexp is analyzed according to ... 

After 20 years. Stern [23] modified these models by including both a compact and a diffuse layer. At the same time, Grahame [24] divided the Stern layer into two regions (i) an inner Helmholtz plane consisting of a layer of adsorbed ions at the surface of the electrode and (ii) an outer Helmholtz plane (referred to as Gouy plane as well), which is formed by the closest approach of diffuse ions to the electrode surface. From the Grahame model, the capacitance C of the double layer is described by Equation 8.1 as follows ... 

The Stern capacitance Ci is not expected to be dependent on the solution composition. Grahame (1947, 1954) assumed C, to be independent on I and deduced its value from the highest NaF concentration available, 0.916 M (Figure 3.12, solid line), by calculating from the GC theory (Equation 3.45) and computing Q... 

Equation [23] represents the GC solution for point ions. A key development in the theory of electrolytes was the introduction of a finite distance of closest approach of ions to a charged surface by Stern and further elaborated upon by Grahame. The layer of ions directly adsorbed onto the surface constitutes the inner Helmholtz layer those ions that make contact but do not adsorb define the abovementioned distance of closest approach and constitute the outer Helmholtz or Stern layer. These modifications still admit an analytical solution to the GC equation Laplace s equation is solved in the Stern layer with the (linear) potential and (constant) field matched at the polyelectrolyte surface and to the outer GC solution. The adsorbed ions serve to reduce the charge density of the surface. Identification of the inner and outer Helmholtz layers has been particularly helpful in improving agreement between GC theory and electrochemical data. If we assign a common radius a to all electrolyte ions, then the identification of the interface atx = a actually... 

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