Double differential capacity

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.)...

Derive the general equation for the differential capacity of the diffuse double layer from the Gouy-Chapman equations. Make a plot of surface charge density tr versus this capacity. Show under what conditions your expressions reduce to the simple Helmholtz formula of Eq. V-17. 

The electrical double layer at Hg, Tl(Ga), In(Ga), and Ga/aliphatic alcohol (MeOH, EtOH) interfaces has been studied by impedance and streaming electrode methods.360,361 In both solvents the value ofis, was independent of cei (0.01 < cucio4 <0.25 M)and v. The Parsons-Zobel plots were linear, with /pz very close to unity. The differential capacity at metal nature, but at a = 0,C,-rises in the order Tl(Ga) < In(Ga) < Ga. Thus, as for other solvents,120 343 the interaction energy of MeOH and EtOH molecules with the surface increases in the given order of metals. The distance of closest approach of solvent molecules and other fundamental characteristics of Ga, In(Ga), Tl(Ga)/MeOH interfaces have been obtained by Emets etal.m... 

Various methods have been employed for the determination of E of liquid and solid metals. Besides purely electrochemical ones (e.g. measurement of the differential double layer capacity (see also chapter 4.2)) further techniques have been used for the investigation of the surface tension at the solid/electrolyte solution phase boundary. The employed methods can be grouped into several families based on the meas-... 

Measurement of the differential capacitance C = d /dE of the electrode/solution interface as a function of the electrode potential E results in a curve representing the influence of E on the value of C. The curves show an absolute minimum at E indicating a maximum in the effective thickness of the double layer as assumed in the simple model of a condenser [39Fru]. C is related to the electrocapillary curve and the surface tension according to C = d y/dE. Certain conditions have to be met in order to allow the measured capacity of the electrochemical double to be identified with the differential capacity (see [69Per]). In dilute electrolyte solutions this is generally the case. 

Measurements of the interfaeial eapacitance (the differential double layer capacity Cdl) have been used widely, the method has been labelled tensammetry [46Bre, 52Bre, 51Dosl, 52Dosl, 63Bre]. Various experimental setups based on arrangements for AC polarography, lock-in-amplifier, impedance measurement etc. have been employed. In all reports evaluated in the lists of data below the authors have apparently taken precautions in order to measure only the value of Cdl-... 

FIG. 8 Inverse differential capacity at the zero surface charge vs. inverse capacity Cj of the diffuse double layer for the water-nitrobenzene (O) and water-1,2-dichloroethane (, ), interface. The diffuse layer capacity was evaluated by the GC ( ) or the MPB (0,)> theory. (From Ref. 22.)... 

Of the quantities connected with the electrical double layer, the interfacial tension y, the potential of the electrocapillary maximum Epzc, the differential capacity C of the double layer and the surface charge density q(m) can be measured directly. The latter quantity can be measured only in extremely pure solutions. The great majority of measurements has been carried out at mercury electrodes. 

Figure 2.9, it can be seen that the interfacial capacitance does show a dependence on concentration, particularly at low concentrations. In addition, whilst there is some evidence of the expected step function away from the pzc, the capacitance is not independent of V. Finally, and most destructive, the Helmholtz model most certainly cannot explain the pronounced minimum in the plot at the pzc at low concentration. The first consequence of Figure 2.9 is that it is no longer correct to consider that differentiating the y vs. V plot twice with respect to V gives the absolute double layer capacitance CH where CH is independent of concentration and potential, and only depends on the radius of the solvated and/or unsolvated ion. This implies that the dy/dK (i.e. straight lines joined at the pzc. Thus, in practice, the experimentally obtained capacitance is (ddifferential capacitance. (The value quoted above of 0.05-0,5 Fm 2 for the double-layer was in terms of differential capacitance.) A particular value of (di M/d V) is obtained, and is valid, only at a particular electrolyte concentration and potential. This admits the experimentally observed dependence of the double layer capacity on V and concentration. All subsequent calculations thus use differential capacitances specific to a particular concentration and potential. 

Figure 7. Comparison of (a, solid) electrochemical and (b, dashed) UHV measurements of the H, coverage/potentiaI differential versus potential on Pt(lll).1.) cathodic sweep (25 mV/s) voltammogram in 0.3 M HF from Ref. 20, constant double layer capacity subtracted, b.) dB/d(A ) versus A plot derived from A versus B plot of Ref. 26. Potential scales aligned at zero coverage. Areas under curves correspond to a.) 0.67 and b.) 0.73 M per surface Pt atom.

At the electrocapillary maximum, the charge density, a, is zero (point of zero charge) (Fig. A.4.5c). By definition, the differential capacity of the double layer, Cd, is equal (Second Lippmann Equation). 

The plots of AG° vs. for differentratios, calculated from the model proposed in Refs. 148, 151, and 152 are presented in Fig.lO. The calculation was made for n = 2, the area occupied by one water molecule equal to 0.09 nm, and for other double-layer parameters that best fit the experimental data on differential capacity of the Hg/water interface. As follows from these plots, no tZmax of adsorption can be reached if the Hfj/n ratio is greater than the dipole moment of water (1.84 D). 

Amokrane and Badiali proposed a semiempirical approach to the determination of the solvent contribution C, to the capacitance of the double layer in aqueous and nonaqueous " solutions. They used the relation C = Cf - C m, where Q is the experimentally determined capacity of the inner layer and Cm is the contribution of the metal. The plots ofC, vs. (Tm were presented for various solvents and correlated with their properties.However, the problem of the supporting electrolyte was entirely neglected in the quoted papers. It was shown recently that the height and position of the maximum on the C, vs. Gm plots depend on the type of the supporting electrolyte. Experimental differential capacity data obtained on the Hg electrode in methanol and ethanol containing various electrolytes with nonadsorbing anions (F , PFg, ClOi) indicate that the type as well as concentration of the electrolyte influences the position and the height of the maximum on the C, vs. plots (Fig. 13). 

Fig. 6-96. Change in differential capacity of an interfadal double layer leading or not leading to interfadal lattice transformation in anodic and cathodic potential sweeps for a gold electrode surface (100) in perchloric add solution Ey = critical potential beyond which the interfadal lattice transforms from (5 x 20) to (1 x 1) E = critical potential below which the interfadal lattice transforms from (1 x 1) to (5 x 20) Ejm = potential of zero charge VacE = volt referred to the saturated calomel electrode. [From Kolb-Schneider, 1985.]...

Naneva and Popov et al. [4, 5] have studied Cd(OOOl) grown electrolytically in a Teflon capillary in NaF aqueous solution. A value of fpzc equal to —0.99 V (versus saturated calomel electrode (SCE)) was evaluated from minimum potential (Amin) on the differential capacity C-E curves obtained in dilute electrolyte. The zero charge potential was found to be practically independent of the crystallographic orientation. The Apzc and the irmer layer capacity of Cd(OOOl) single crystals were determined in KF solution as a function of temperature [5]. The positive values of AApzc/AT indicated that the water dipoles in the inner part of the double layer were orientated with their negative part to the electrode surface. It was found that the hydrophilicity of the electrodes was increasing in the order Cd(OOOl) < Ag(100)[Pg.768]

The differences between various Ag surfaces can be distinguished by comparing their surface morphology (generally, the surface of (110) crystal is more folded than that of (111)) and other properties, such as the surface density of atoms, the PZC, and double-layer capacitance. The double-layer properties of single-crystal Ag electrodes have been studied very intensively [3, 22-27]. Selected characteristics of various Ag surfaces are compared in Table 1, which shows that the higher the surface density of atoms, the more positive PZC becomes. Furthermore, Fig. 2 exemplifies differential capacity data of those Ag surfaces. 

What is the significance of this equation It shows that the slope of the curve of the electrode charge versus cell potential yields the value of the differential capacity of the double layer. In the case of an ideal parabolic y versus Vcurve, which yields a linear q versus Vcurve, one obtains a constant capacitance (Fig. 6.56). 

Going a step further, what does the parallel-plate model of the double layer have to say regarding the capacity of the interface Rearranging Eq. (6.119) in the form of the definition of differential capacity [Eq. (6.97)],... 

It appears that an electrified interface does not behave like a simple double layer. The parallel-plate condenser model is too naive an approach. Evidently some crucial secrets about electrified interfaces are contained in those asymmetric electrocapillaiy curves and the differential capacities that vary with potential. One has to think again. 

It was seen in Section 6.6.6 that the separation of charges and potentials in the double-layer region was done so that the total differential capacity could be divided into two capacities in series,... 

Section 6.7.7 ended with an encouraging statement The contribution of the water dipoles constituting the hydration sheath of the electrode can be ignored in the understanding of the differential capacity of the electrified interface. Thus, if water molecules, in spite of their large number in the interfacial region, are not responsible for this property of the double layer, what is We also said that the total differential capacity of the interface could be divided into two contributions... 

In the absence of specific adsorption and dipolar contributions, there is no excess charge in the whole double layer when positive and negative ions are equally distributed at the plane of closest approach qM and A02 will be both zero. The corresponding electrode potential is the potential of zero charge (pzc) which can be evaluated from the minimum in the differential capacity—potential curve for a metal electrode in contact with a dilute electrolyte [6]... 

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 any event, there is always a strict, equilibrium relation between the charge density, qM, on the electrode sur ce and the total potential difference, E, between the bulk phases of electrode and solution. This relation is often characterized by the differential double-layer capacity, Cd, defined as... 

The current density, will be the sum of the faradaic current density, jF, and the charging current density, c, cf. eqn. (8). The latter is related to the interfacial potential indicated in an implicit way by eqn. (20). The theory of the electrical double layer provides no analytical expression for the relation between E and qM and so, rigorously, this part of the problem would have to be solved numerically using the empirical relationship, which is known for many commonly used indifferent electrolytes. If Cd = dqM/dE is the differential double-layer capacity, we have... 

Fig. 4. Differential double-layer capacity as a function of d.c. potential of the mercury electrode in aqueous solutions of 0.1 M potassium salts, From ref. 20.

In its most simple form, this means without effects such as adsorption or formation of coatings at the electrode surface36. The resistance, Rc, represents electrical conductivity of the electrolyte and is not a property of the electrode itself. The differential double-layer capacity, Cmetal surface of the metal-electrolyte interface, which is in equilibrium with an equal excess of charge but opposite in sign at the side of the electrolyte. 

Conversion of values of y into capacities is done by double differentiation in relation to the electrostatic potential difference A0 between its value in the metal, 0M, and that in the solution, (j)s. The first derivative gives the charge on the interface, and is the Lippmann equation... 

Fig. 3.3 Schematic plots of the double layer region, (a) Electrocapillary curve (surface tension, y, vs. potential) (b) Charge density on the electrode, aM, vs. potential (c) Differential capacity, Cd, vs. potential. Curve (b) is obtained by differentiating curve (a), and (c) by differentiation of (b), Ez is the point of zero...

Figure 3.4 gives examples of real electrocapillary curves and differential capacity curves. Double layer models have to explain the shape of these curves. 

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