Double capacitance

Figure Bl.28.8. Equivalent circuit for a tliree-electrode electrochemical cell. WE, CE and RE represent the working, counter and reference electrodes is the solution resistance, the uncompensated resistance, R the charge-transfer resistance, R the resistance of the reference electrode, the double-layer capacitance and the parasitic loss to tire ground.

Electrically, the electrical double layer may be viewed as a capacitor with the charges separated by a distance of the order of molecular dimensions. The measured capacitance ranges from about two to several hundred microfarads per square centimeter depending on the stmcture of the double layer, the potential, and the composition of the electrode materials. Figure 4 illustrates the behavior of the capacitance and potential for a mercury electrode where the double layer capacitance is about 16 p.F/cm when cations occupy the OHP and about 38 p.F/cm when anions occupy the IHP. The behavior of other electrode materials is judged to be similar. 

Fig. 7. (a) Simple battery circuit diagram where represents the capacitance of the electrical double layer at the electrode—solution interface, W depicts the Warburg impedance for diffusion processes, and R is internal resistance and (b) the corresponding Argand diagram of the behavior of impedance with frequency, for an idealized battery system, where the characteristic behavior of A, ohmic B, activation and C, diffusion or concentration (Warburg... 

From an electroanalytical point of view, the double-layer capacitance is a nuisance resulting in the charging current, which has no analytical value. 

Active electrochemical techniques are not confined to pulse and linear sweep waveforms, which are considered large ampHtude methods. A-C voltammetry, considered a small ampHtude method because an alternating voltage <10 mV is appHed to actively couple through the double-layer capacitance, can also be used (15). An excellent source of additional information concerning active electroanalytical techniques can be found in References 16—18. Reference 18, although directed toward clinical chemistry and medicine, also contains an excellent review of electroanalytical techniques (see also... 

Even in the absence of Faradaic current, ie, in the case of an ideally polarizable electrode, changing the potential of the electrode causes a transient current to flow, charging the double layer. The metal may have an excess charge near its surface to balance the charge of the specifically adsorbed ions. These two planes of charge separated by a small distance are analogous to a capacitor. Thus the electrode is analogous to a double-layer capacitance in parallel with a kinetic resistance. 

Electrode surfaces in elec trolytes generally possess a surface charge that is balanced by an ion accumulation in the adjacent solution, thus making the system electrically neutral. The first component is a double layer created by a charge difference between the electrode surface and the adjacent molecular layer in the flmd. Electrode surfaces may behave at any given frequency as a network of resistive and capacitive elements from which an elec trical impedance may be measured and analyzed. 

The model just presented describes what electrochemists call the diffuse part of the double layer and no account is made of the inner layer effects such as the plane of the closest approach. To have an idea what the impact of the effects predicted by this model on the measured capacitance could be, we assume the traditional inner and diffuse layer separation. However, we... 

The capacitance. The electrical double layer may be regarded as a resistance and capacitance in parallel see Section 20.1), and measurements of the electrical impedance by the imposition of an alternating potential of known frequency can provide information on the nature of a surface. Electrochemical impedance spectroscopy is now well established as a powerful technique for investigating electrochemical and corrosion systems. 

It is possible on the basis of this model (arrangement O) to explain the constant capacitance region on the negative side of the C vs. E curve (Fig. 20.7), and why the capacitance in this region is independent of the nature of the cations in the solution. The model of the double layer is shown in Fig. 20.12 in which it can be seen that the surface of the electrode and the... 

Fig. 20.12 Double layer consisting of (a) a layer of water dipoles and a layer of ions that may be regarded as (b and c) two capacitors in series (after Bockris and Reddy ). Note that A rH are the capacitances of the regions of high and low permittivities, respectively...

The capacitance of the double layer consists of combination of the capacitance of die compact layer in series with that of the diffuse layer. For two capacitors in series, the total capacitance is given by... 

Figure 1-13 displays the experimental dependence of the double-layer capacitance upon the applied potential and electrolyte concentration. As expected for the parallel-plate model, the capacitance is nearly independent of the potential or concentration over several hundreds of millivolts. Nevertheless, a sharp dip in the capacitance is observed (around —0.5 V i.e., the Ep/C) with dilute solutions, reflecting the contribution of the diffuse layer. Comparison of the double layer witii die parallel-plate capacitor is dius most appropriate at high electrolyte concentrations (i.e., when C CH). 

The charging of the double layer is responsible for the background (residual) current known as the charging current, which limits die detectability of controlled-potential techniques. Such a charging process is nonfaradaic because electrons are not transferred across the electrode-solution interface. It occurs when a potential is applied across the double layer, or when die electrode area or capacitances are changing. Note that the current is the tune derivative of die charge. Hence, when such processes occur, a residual current flows based on die differential equation... 

FIGURE 1-13 Double-layer capacitance of a mercury drop electrode in NaF solutions of different concentrations. (Reproduced with permission from reference 5.)... 

Measurements of the double-layer capacitance provide valuable insights into adsorption and desorption processes, as well as into the structure of film-modified electrodes (6). 

The more highly charged the interface becomes, the more the charges repel each other, thereby decreasing the cohesive forces, lowering the surface tension, and flattening the mercury drop. The second differential of the electrocapillary plot gives directly the differential capacitance of the double layer ... 

For (ideally) polarizable metals with a sufficiently broad double-layer region, such as Hg, Ag, Au, Bi, Sn, Pb, Cd, H, and others, Ea=to can be obtained from measurements of the double-layer capacitance in dilute... 

The model more generally accepted for metal/electrolyte interfaces envisages the electrical double layer as split into two parts the inner layer and the diffuse layer, which can be represented by two capacitances in series.1,3-7,10,15,32 Thus, the total differential capacitance C is equal to... 

Measurements based on the Gouy-Chapman-Stem theory to determine the diffuse double-layer capacitance 10, 24,72, 74... 

In the second group of models, the pc surface consists only of very small crystallites with a linear parameter y, whose sizes are comparable with the electrical double-layer parameters, i.e., with the effective Debye screening length in the bulk of the diffuse layer near the face j.262,263 In the case of such electrodes, inner layers at different monocrystalline areas are considered to be independent, but the diffuse layer is common for the entire surface of a pc electrode and depends on the average charge density <7pc = R ZjOjOj [Fig. 10(b)]. The capacitance Cj al is obtained by the equation... 

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