The Adsorption Pseudocapacitance

The adsorption isotherms discussed in Section. 11.1 describe the potential dependence of the fractional surface coverage, 0. For intermediates formed in a charge-transfer process, (cf. Eq. (11.7)), the fractional coverage is associated with a Faradaic charge gp that is given by 

the adsorption isotherm also yields the dependence of the Faradaic charge consumed in forming the adsorbed intermediate on potential. This allows us to define a new type of capacitance, which is called the adsorption pseudocapacitance, C j,  

The capacitive nature of the adsorption pseudocapacitance can be further illustrated by considering its response to an alternating voltage (AC) perturbation. Let us assume that a low-amplitude sinusoidal voltage signal is applied to a system at equilibrium. The sinusoidal waveform can be expressed by the equation 

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The adsorption pseudocapacitance can be readily calculated from the appropriate isotherm, with the use of its definition, given in Eq. 391. This is shown next for the Langmuir and Frumkin isotherms. 

We proceed here to derive the expressions for the adsorption pseudocapacitance under Frumkin conditions, which we shall denote C. ... 

There is an apparent discrepancy between the treatment of electrode kinetics under Temkin conditions, at intermediate values of the coverage, and the results shown in Fig. 141(b) for the adsorption pseudocapacitance in the same region. For the purpose of calculating the kinetic parameters, we have assumed that 0 is a linear function of potential. This is a valid assumption, as we can see in Fig. 21. Yet such a linear dependence of 6 on should give rise to a constant value... 

The dependence of the impedance associated with on frequency can be visualized with the aid of the equivalent circuit shown in Fig. 121(b). Here the adsorption pseudocapacitance is part of a more complete circuit, showing the other circuit elements we have already... 

Since, by definition, 0 < 0 < 1, Eq. 591 has physically meaningful solutions only for / < - 4. A plot of the Frumkin isotherm for negative values of the parameter / is shown in Fig. 171. For / = - 12, the solution of Eq. 591 yields 0 = 0.11 and 0.89. Between these values, the coverage appears to increase with decreasing potential, which would imply a negative value of the adsorption pseudocapacitance. This does not represent physical reality, of course. If we trace the potential in the positive direction, 0 will Jump from 0.11 to 0.999. When the... 

Another complication with the use of this treatment must be noted it lies in the nature of the pseudocapacitance quantity used. The adsorption pseudocapacitance is defined as the product of the charge density for monolayer coverage, ij, and the derivative of coverage with potential, Eq. (45) ... 

In the more interesting case here where 6 is significant and potential dependent, Qi must be replaced, to an approximation, by Qi + where is the adsorption pseudocapacitance of the chemisorbed intermediate derived from differentiating the (Langmuir) isotherm... 

The adsorption pseudocapacitance is dominated by the term ddjdE and hence a plot of Cq versus-S gives information about the coverage directly. Figure 1.15 shows a set of Cs-E plots for a pentanol solution at a mercury electrode. The peaks are due to adsorption/desorption processes, so, for example, from the 0.1 M solution, the alcohol adsorbs in the potential range between —0.1 V and —1.1 V. 

This kinetic parameter is inversely proportional to the rate constant of the electrode process of adsorption. On the other hand, the adsorption pseudocapacitance reduces to... 

Adsorption kinetic. (A) Small deviations from equilibrium-frequency dependence. At small deviations from equilibrium, the value of the adsorption pseudocapacitance. Cad (see Eq. (16)), when measured by the a.c. method is the function of the frequency decreasing from the equilibrium value at very low frequency to zero at high frequencies. The... 

Note that the resistance R,j> is an integral part of the physical phenomenon that gives rise to the formation of the adsorption pseudocapacitance. It is a Faradaic resistance, since C< > is due to a charge-transfer process. The association of this charge-transfer process with the formation of an adsorbed intermediate, which can proceed only xmtil the appropriate coverage has been reached, is manifested by placing the resistor in series with the capacitor. It should also be borne in mind that both Cequivalent circuits representing the electrochemical interface... 

Figure 11.4 The equivalent circuits for (a) the adsorption pseudocapacitance and the corresponding resistance / <, and (b) an interface containing an adsorption pseudocapacitance.

The dependence of Cadsorption pseudocapacitance on potential is shown in Figure 11.5b. This... 

Thus, the current is proportional to the sweep rate, and its variation with potential represents the dependence of the adsorption pseudocapacitance on potential, as discussed in Section 11.2. Although the current observed increases linearly with sweep rate, the total charge needed to form a UPD layer is independent of it. 

In Section 11.2 we discussed the concept of the adsorption pseudocapacitance and its dependence on potential and the fractional coverage. The phenomenon of underpotential deposition is discussed in Chapter 12, noting that UPD is a prime example in which the adsorption pseudocapacitance plays a role. Both phenomena are studied in most cases (but not exclusively) by applying cyclic voltammetry. Here we discuss the theory behind cyclic voltammetry associated with the above two phenomena... 

In another study of DMFC anodes, shovm in Figure 16.10, the complex-plane impedance plots were studied as a function of the current density applied. The diameters of the semicircles were found to decrease with increasing current density, as expected, but the new feature observed is an inductive branch of the curves. This can be modeled, of course, by adding an inductive element to the equivalent circuit representation, in series with the Faradaic resistance, but the physical origin of this added circuit element is still open for debate. There is a tendency to associate it with sluggish adsorption of CO, formed as an intermediate in the oxidation of methanol. However, unlike the adsorption pseudocapacitance, which is well understood (cf Section 11.2), there is no theory for the dependence of the pseudoinductance on potential, coverage or any other measured parameter. 

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