Double integral capacity

The shape of the electrocapillary curve is easily calculated if it is assumed that the double layer acts as a condenser of constant capacity C. In this case, double integration of Eq. V-50 gives... 

Fig. V-12. Variation of the integral capacity of the double layer with potential for 1 N sodium sulfate , from differential capacity measurements 0, from the electrocapillary curves O, from direct measurements. (From Ref. 113.)...

Cd differential capacity of double layer Cj integral capacity of double layer Cs capacity in RC series combination... 

Experiments show that the values for the double layer capacity at single crystal metal surfaces depend on the nature of the metal. This indicates that the metal surface cannot be considered as a perfect conductor, as was done in classical theories. It is well known that an overspill of metal electrons can occur at the boundary of a metal with vacuum. A similar overspill expected at a metal-solution interface would alter the double layer capacity by an amount depending on the type of metal. Models have been constructed in which the metal is represented by an electronic plasma in a uniformly, positively charged background, which is known as the jellium model [81,82]. The inclusion of the electron overspill into the integral equation theories of the electric double layer has been performed basically with the HAB model. 

In applying the Debye formulae to the calculation of heat capacities we require values of the double integral... 

Area, constant, optical absorption Activity, absorption coefficient Debye length Cyclic voltammogram Capacitance / j,F Double layer capacity / 0.F Differential double layer capacity Integral double layer capacity Concentration / M Surface concentration Bulk concentration / M Diffusion coefficient / cm s ... 

The experimental methods available for determination of the surface concentration of adsorbed intermediates formed in charge-transfer processes have been reviewed recently by Gileadi and Conway. They have been divided into two groups (a) methods in which the charge needed to form or desorb the surface species is directly measured and b) methods in which the capacity at the interface (i.e., the sum of the ionic double-layer capacity and the adsorption pseudocapacity) is measured as a function of potential and integrated to obtain the charge. These methods will not be discussed further here. The experimental methods described below pertain to the determination of adsorption of uncharged species on solid electrodes. 

Therefore from the basic experimental data such as a state equation or thermoelastic coefficients, heat capacities etc., it is possible to obtain thermodynamic coefficients. A double integration of equation [2.25] will give a characteristic function, i.e. knowing the complete thermodynamics of the phase studied. 

One of the properties of the double layer that is its difterential or integral capacity... 

The concordance between the three different methods is very satisfactory as is illustrated in Fig. 2, where the integral capacity of the double layer mercury... 

Electrochemical techniques such as chronocoulometry, integrated cyclic voltammetry curves, or differentiating double-layer capacity measurements as a function of bulk concentrations can be used to estimate FA however, these types of determinations may still be in error since they measure the total number of surface species, while the number of molecules at active sites which exhibit both the EM and CT effects may be much smaller. Thus, the estimates of Gsers 10 to 10 found at pretreated Ag, for example, may err on the low side if special active sites are involved in SERS. It has been estimated that only 3% of the surface sites are SERS active. " ... 

If qM is evaluated experimentally, e.g. from the integration of differential capacity curves, A02 can be calculated using eqn. (44) of the diffuse double layer theory. Figure 3 shows the variation of A02 with... 

In contrast to the chronocoulometric technique, the information obtained from the impedance technique concerns the derivatives with respect to potential of To, In /30 and d(ln (30)/dE. It is therefore needed to postulate an explicit isotherm in order to fit experimental data to the theoretical equations. However, the results, e.g. for (dT0/dE, can be integrated to give T0 as a function of potential within the faradaic region. As chronocoulometry is, by definition, applicable to potentials outside the faradaic region, the two methods may be considered as complementary tools to study adsorption phenomena that are hard to detect in the classical way, i.e. from double-layer data (e.g. the capacity Cd) that are influenced by the presence of electro-inactive absorbates. 

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