Measured capacitance

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. 

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 is a readily measured interfacial property and it gives qualitative information on the adsorption of species at the electrode surface. Since the surface charge density, q, is a function of the potential and of coverage, the measured capacitance may be expressed as the sum of a true (high frequency) capacitance and an adsorption pseudocapacitance, i.e. q f(E,6) and hence... 

Calculations of the capacitance of the mercury/aqueous electrolyte interface near the point of zero charge were performed103 with all hard-sphere diameters taken as 3 A. The results, for various electrolyte concentrations, agreed well with measured capacitances as shown in Table 3. They are a great improvement over what one gets104 when the metal is represented as ideal, i.e., a perfectly conducting hard wall. The temperature dependence of the compact-layer capacitance was also reproduced by these calculations. 

Figure 22. Steady-state polarization curves of aluminum in pure and mixed NaOH -f NaCl solutions , 4Af NaOH A, 4 M NaOH + 2 M NaCl O, 1 M NaOH + 2 M NaCl , 2 M NaCl (pH 1 to 13). Labels on the lines denote measured capacitances of the interface.

We now discuss some of the experimental aspects of temperature spectroscopy. Lang (1974) called his original method deep level transient spectroscopy (DLTS), and he measured capacitance transients produced by voltage pulses in diodes made from conductive materials. However, in SI materials, this method is not feasible and an alternate method, involving current transients produced by light pulses in bulk material (or Schottky structures), was... 

In many cases we have an idea about the number of charged groups on a surface. Then we might want to know the potential. The question is how are surface charge a and surface potential V o related This question is also important because if we know a fo) we can calculate da/dtpo- This is basically the capacitance of the double layer and can be measured. The measured capacitance can be compared to the theoretical result to verify the whole theory. 

From the above equation, the measured capacitance between the source and the detector electrodes of the electrode pair i is determined from the given dielectric constant (permittivity) distribution of the medium under investigation. The processes of finding the capacitance for a given permittivity distribution is referred to as the forward problem. On the other hand, the process of finding the permittivity distribution from a set of capacitance measurements is referred to as the inverse problem. 

Giant axons from squid have a large diameter ranging from 300 to 700 ym. Because of this large size, we are able to insert metal electrodes directly into the axon and measure capacitance and conductance across the membrane. This is the most unequivocal method to measure transmembrane capacitance and is far better than the use of external electrodes as done previously by Cole and Curtis (11). In spite of the simplicity and ease of this technique, there are still a few unsolved problems which will be discussed later. Figure 3 shows one of the exemplary results of nerve membrane capacitance and conductance measurements. Comparison of this result with the one shown in Figure 2 readily demonstrates that there are considerable differences between these two sets of curves. 

In an attempt to rationalize the measured capacitance values, and especially the low value for the basal plane (ca. 3pF/cm2), these authors first concluded that space charge within the electrode is the dominant contribution (rather than the compact double layer with ca. 15-20 pF/cm2, or the diffuse double layer with >100 pF/cm2). They then applied the theory of semiconductor electrodes to confirm this and obtained a good agreement by assuming for SAPG a charge carrier density of 6 x 1018/cm3 and a dielectric constant of 3 for GC, they obtained 13 pF/cm2 with the same dielectric constant and 1019 carriers per cubic centimeter. 

Yet, apart from the purely quantitative criterion, one should also take into consideration general reasoning. When comparing the discussed methods, we note that the er 2 vs. E plot is frequency-independent indeed, this is because a is frequency-independent by definition [see Eq. (2)]. This plot is believed to be the most universal for comparing results of different experiments because the contribution, coming from the frequency-dependent factor proper, is separated in this case. Unfortunately, the dimensionality of a precludes a correct comparison of the results thus computed with the directly measured capacitance values published in the literature. 

The interesting part of this capacitance is that which is due to the presence of the material between the plates. Write C0 for the relatively uninteresting case in which the plates are separated only by vacuum. The dielectric susceptibility e(a>) of the intervening substance is defined as the ratio of the measured capacitance C(a>) compared with Q ... 

Figure 5.13 (top) displays the frequency spectra of the measured capacitance for temperatures ranging from 20 K to 300 K for a standard cell (ITO/PEDOT/MDMO-PPV PCBM/Al). The arrow indicates increasing temperatures. One clearly observes a step which is shifted to higher frequencies as the temperature increases. In order to evaluate the position of the steps, it is better to plot wdC/dw versus w, rather than C(u>) versus w. Figure 5.13 (bottom) shows the normalised deviated frequency spectrum of the capacitance. The steps now appear as maxima within the individual curves, and the corresponding critical frequency wq can be derived more ac-... 

Measurement leads used to connect to the electrochemical cell contribute measured capacitance. Lead capacitance between the potentiostat and the electrometer is normally accounted for in the potentiostat calibration, but stray capacitance associated with the leads and connections to the cell may not. Moreover, this capacitance may change with time the leads age, and the connections become oxidized. Uncompensated capacitance may also exist within the measurement circuitry of the potentiostat itself. These uncompensated capacitances impose a practical lower limit on the coating capacitance that can be measured. 

Measurement of calibrated capacitors can also be used to determine instrument limitations. Figure 36 shows a plot of the variation in measured capacitance versus known capacitance for a commercial impedance analyzer system (110). 

Interfacial states may also be explored. Assuming that these states are slow and do not contribute to the measured capacitance (which can obviously be ensured, in principle, by working at a high enough frequency), they may be detected through the effects of ionisation on the potential distribution. It is easily seen that the change in surface occupancy, Ant (cm 2), is given by... 

An important and precise traditional method of measuring capacitance for dipole moment determinations is the heterodyne beat method, a particular form of the null method mentioned above. The output of an LC oscillator incorporating the capacitance... 

Another type of electronic circuit, the relaxation oscillator, can be used to measure capacitance a simple apparatus for this purpose has been described by Kurtz, Anderson, and Willeford. ... 

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