Differential capacities

It should be noted that the capacity as given by C, = a/E, where a is obtained from the current flow at the dropping electrode or from Eq. V-49, is an integral capacity (E is the potential relative to the electrocapillary maximum (ecm), and an assumption is involved here in identifying this with the potential difference across the interface). The differential capacity C given by Eq. V-50 is also then given by... 

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.)...

The variation of the integral capacity with E is illustrated in Fig. V-12, as determined both by surface tension and by direct capacitance measurements the agreement confrrms the general correctness of the thermodynamic relationships. The differential capacity C shows a general decrease as E is made more negative but may include maxima and minima the case of nonelectrolytes is mentioned in the next subsection. 

Evidence for two-dimensional condensation at the water-Hg interface is reviewed by de Levie [135]. Adsorption may also be studied via differential capacity data where the interface is modeled as parallel capacitors, one for the Hg-solvent interface and another for the Hg-adsorbate interface [136, 137]. 

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 above formulas combined with Eqs. (74) and (75) taken at zero charge density yield Eq. (54) for the differential capacitance. Eq. (82) can be used recursively to generate the derivatives of the differential capacity at zero charge density to an arbitrary order, though the calculations become rather tedious already for the second derivative. Thus, in principle at least, we can develop capacitance in the Taylor series around the zero charge density. The calculations show that the capacitance exhibits an extremum at the point of zero charge only in the case of symmetrical ions, as expected. In contrast with the NLGC theory, this extremum can be a maximum for some values of the parameters. In the case of symmetrical ions the capacitance is maximum if + — a + a, < 1. We can understand this result... 

These electrodes have been evaluated in lithium-polymer cells that operate at 85 °C [35]. Although the electrochemical discharge is not yet fully understood, differential capacity plots have shown evidence of two reversible ordering transitions and a kinetically slow phase transformation. 

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... 

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. 

In vivo and in vitro studies on the differentiation and proliferation of immature rat thymus subsets have shown that dibutyltin dichloride reduces the production of CD4 CD8 and mature single-positive thymocyte proliferation by selectively inhibiting immature CD4 CD8 thymocyte proliferation but without affecting the differentiation capacity of these cells, suggesting that thymocyte proliferation and differentiation are separately regulated processes (Pieters et al., 1993, 1994a,b, 1995). 

The structure of the interface between two immiscible electrolyte solutions (ITIES) has been the matter of considerable interest since the beginning of the last century [1], Typically, such a system consists of water (w) and an organic solvent (o) immiscible with it, each containing an electrolyte. Much information about the ITIES has been gained by application of techniques that involve measurements of the macroscopic properties, such as surface tension or differential capacity. The analysis of these properties in terms of various microscopic models has allowed us to draw some conclusions about the distribution and orientation of ions and neutral molecules at the ITIES. The purpose of the present chapter is to summarize the key results in this field. 

The closely related quantity is the differential capacity C, which is defined by ... 

In particular, the coupling between the ion transfer and ion adsorption process has serious consequences for the evaluation of the differential capacity or the kinetic parameters from the impedance data [55]. This is the case, e.g., of the interface between two immiscible electrolyte solutions each containing a transferable ion, which adsorbs specifically on both sides of the interface. In general, the separation of the real and the imaginary terms in the complex impedance of such an ITIES is not straightforward, and the interpretation of the impedance in terms of the Randles-type equivalent circuit is not appropriate [54]. More transparent expressions are obtained when the effect of either the potential difference or the ion concentration on the specific ion adsorption is negli-... 

Samec et al. [15] used the AC polarographic method to study the potential dependence of the differential capacity of the ideally polarized water-nitrobenzene interface at various concentrations of the aqueous (LiCl) and the organic solvent (tetrabutylammonium tetra-phenylborate) electrolytes. The capacity showed a single minimum at an interfacial potential difference, which is close to that for the electrocapillary maximum. The experimental capacity was found to agree well with the capacity calculated from Eq. (28) for 1 /C,- = 0 and for the capacities of the space charge regions calculated using the GC theory,... 

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.)... 

FIG. 9 Differential capacity C of the interface between 0.1 M LiCl in water and 0.02 M tetrabu-tylammonium tetraphenylborate ( ) or tetrapentylammonium tetrakis[3,5-bis(trifluoromethyl)phe-nyl]borate ( ) in o-nitrophenyl octyl ether as a function of the interfacial potential difference (From Ref 73.)... 

There is a group of substances, in the presence of which significant changes in the surface tension of the ITIES were observed, which are also likely to influence the differential capacity of the ITIES correspondingly. These substances include various ionic and nonionic surfactants (Section IV.B.2) and amphiphilic phospholipids (Section IV.B.3) or affinity dyes. Attention has focused on phospholipids. 

Wu CY, Kirman JR, Rotte MJ, et al. Distinct lineages of T(H)1 cells have differential capacities for memory cell generation in vivo. Nat Immunol 2002 3 852-858. 

The second differential of the electrocapillary curve yields the differential capacity of the electrode, C ... 

The differential capacity is given by the slope of the tangent to the curve of the dependence of the electrode charge on the potential, while the integral capacity at a certain point on this dependence is given by the slope of the radius vector of this point drawn from the point Ep = pzc. 

The zero-charge potential is determined by a number of methods (see Section 4.4). A general procedure is the determination of the differential capacity minimum which, at low electrolyte concentration, coincides with Epzc (Section 4.3.1). With liquid metals (Hg, Ga, amalgams, metals in melts) Epzc is directly found from the electrocapillary curve. 

In this manner, the surface excess of ions can be found from the experimental values of the interfacial tension determined for a number of electrolyte concentrations. These measurements require high precision and are often experimentally difficult. Thus, it is preferable to determine the surface excess from the dependence of the differential capacity on the concentration. By differentiating Eq. (4.2.30) with respect to EA and using Eqs (4.2.24) and (4.2.25) in turn we obtain the Gibbs-Lippmann equation... 

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