Controlled current

In controlled current experiments, the gradient g is known at any one time and Cq can be expressed by Eq. 4.93 above. For n 5, this is 

Exactly how this is used, depends upon the simulation technique. In the explicit method, one will use, e.g., Eq. 4.93, to compute c from present c, . .. -1 then recompute the whole concentration profile by the diffusion calculation. As will be seen (Chapt. 5), a variant of the Crank-Nicolson method correctly assumes 

The electrode potential calculations, after Cq is computed, do not of course differ from those presented in Sects. 4.2.1 and 4.2.2. 

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The operator sets TU on the area to be controlled and puts the button B Under the command of SC EDC impulse of proper value and duration is formed in source controllable current (SCC). 

The largest division of interfacial electrochemical methods is the group of dynamic methods, in which current flows and concentrations change as the result of a redox reaction. Dynamic methods are further subdivided by whether we choose to control the current or the potential. In controlled-current coulometry, which is covered in Section IIC, we completely oxidize or reduce the analyte by passing a fixed current through the analytical solution. Controlled-potential methods are subdivided further into controlled-potential coulometry and amperometry, in which a constant potential is applied during the analysis, and voltammetry, in which the potential is systematically varied. Controlled-potential coulometry is discussed in Section IIC, and amperometry and voltammetry are discussed in Section IID. 

Coulometric methods of analysis are based on an exhaustive electrolysis of the analyte. By exhaustive we mean that the analyte is quantitatively oxidized or reduced at the working electrode or reacts quantitatively with a reagent generated at the working electrode. There are two forms of coulometry controlled-potential coulometry, in which a constant potential is applied to the electrochemical cell, and controlled-current coulometry, in which a constant current is passed through the electrochemical cell. 

A second approach to coulometry is to use a constant current in place of a constant potential (Figure 11.23). Controlled-current coulometry, also known as amperostatic coulometry or coulometric titrimetry, has two advantages over controlled-potential coulometry. First, using a constant current makes for a more rapid analysis since the current does not decrease over time. Thus, a typical analysis time for controlled-current coulometry is less than 10 min, as opposed to approximately 30-60 min for controlled-potential coulometry. Second, with a constant current the total charge is simply the product of current and time (equation 11.24). A method for integrating the current-time curve, therefore, is not necessary. 

Current-time curve for controlled-current coulometry. 

Ladder diagrams for the controlled-current coulometric analysis of Fe + (a) without the addition of Ce +, and (b) with the addition of Ce +. The matrix is 1 M H2SO4 in both cases. 

End Point Determination Adding a mediator solves the problem of maintaining 100% current efficiency, but does not solve the problem of determining when the analyte s electrolysis is complete. Using the same example, once all the Fe + has been oxidized current continues to flow as a result of the oxidation of Ce + and, eventually, the oxidation of 1T20. What is needed is a means of indicating when the oxidation of Fe + is complete. In this respect it is convenient to treat a controlled-current coulometric analysis as if electrolysis of the analyte occurs only as a result of its reaction with the mediator. A reaction between an analyte and a mediator, such as that shown in reaction 11.31, is identical to that encountered in a redox titration. Thus, the same end points that are used in redox titrimetry (see Chapter 9), such as visual indicators, and potentiometric and conductometric measurements, may be used to signal the end point of a controlled-current coulometric analysis. For example, ferroin may be used to provide a visual end point for the Ce -mediated coulometric analysis for Fe +. 

Instrumentation Controlled-current coulometry normally is carried out using a galvanostat and an electrochemical cell consisting of a working electrode and a counterelectrode. The working electrode, which often is constructed from Pt, is also... 

The other necessary instrumental component for controlled-current coulometry is an accurate clock for measuring the electrolysis time, fe, and a switch for starting and stopping the electrolysis. Analog clocks can read time to the nearest +0.01 s, but the need to frequently stop and start the electrolysis near the end point leads to a net uncertainty of +0.1 s. Digital clocks provide a more accurate measurement of time, with errors of+1 ms being possible. The switch must control the flow of current and the clock, so that an accurate determination of the electrolysis time is possible. 

Coulometric Titrations Controlled-current coulometric methods commonly are called coulometric titrations because of their similarity to conventional titrations. We already have noted, in discussing the controlled-current coulometric determination of Fe +, that the oxidation of Fe + by Ce + is identical to the reaction used in a redox titration. Other similarities between the two techniques also exist. Combining equations 11.23 and 11.24 and solving for the moles of analyte gives... 

Coulometry may be used for the quantitative analysis of both inorganic and organic compounds. Examples of controlled-potential and controlled-current coulometric methods are discussed in the following sections. 

Controllcd-Currcnt Coulomctry The use of a mediator makes controlled-current coulometry a more versatile analytical method than controlled-potential coulome-try. For example, the direct oxidation or reduction of a protein at the working electrode in controlled-potential coulometry is difficult if the protein s active redox site lies deep within its structure. The controlled-current coulometric analysis of the protein is made possible, however, by coupling its oxidation or reduction to a mediator that is reduced or oxidized at the working electrode. Controlled-current coulometric methods have been developed for many of the same analytes that may be determined by conventional redox titrimetry. These methods, several of which are summarized in Table 11.9, also are called coulometric redox titrations. 

Representative Method Every controlled-potential or controlled-current coulo-metric method has its own unique considerations. Nevertheless, the following procedure for the determination of dichromate by a coulometric redox titration provides an instructive example. 

Scale of Operation Coulometric methods of analysis can be used to analyze small absolute amounts of analyte. In controlled-current coulometry, for example, the moles of analyte consumed during an exhaustive electrolysis is given by equation 11.32. An electrolysis carried out with a constant current of 100 pA for 100 s, therefore, consumes only 1 X 10 mol of analyte if = 1. For an analyte with a molecular weight of 100 g/mol, 1 X 10 mol corresponds to only 10 pg. The concentration of analyte in the electrochemical cell, however, must be sufficient to allow an accurate determination of the end point. When using visual end points, coulometric titrations require solution concentrations greater than 10 M and, as with conventional titrations, are limited to major and minor analytes. A coulometric titration to a preset potentiometric end point is feasible even with solution concentrations of 10 M, making possible the analysis of trace analytes. 

Accuracy The accuracy of a controlled-current coulometric method of analysis is determined by the current efficiency, the accuracy with which current and time can be measured, and the accuracy of the end point. With modern instrumentation the maximum measurement error for current is about +0.01%, and that for time is approximately +0.1%. The maximum end point error for a coulometric titration is at least as good as that for conventional titrations and is often better when using small quantities of reagents. Taken together, these measurement errors suggest that accuracies of 0.1-0.3% are feasible. The limiting factor in many analyses, therefore, is current efficiency. Fortunately current efficiencies of greater than 99.5% are obtained routinely and often exceed 99.9%. 

Precision Precision is determined by the uncertainties of measuring current, time, and the end point in controlled-current coulometry and of measuring charge in controlled-potential coulometry. Precisions of +0.1-0.3% are routinely obtained for coulometric titrations, and precisions of +0.5% are typical for controlled-potential coulometry. 

Time, Cost, and Equipment Controlled-potential coulometry is a relatively time-consuming analysis, with a typical analysis requiring 30-60 min. Coulometric titrations, on the other hand, require only a few minutes and are easily adapted for automated analysis. Commercial instrumentation for both controlled-potential and controlled-current coulometry is available and is relatively inexpensive. Low-cost potentiostats and constant-current sources are available for less than 1000. 

Apparatus and circurtry associated with controlled current measurements. 

Speed potentiometer Speed comparator Speed amplifier and controller Current comparator Current amplifier and controller... 

For control cables, derating may not be material in view of very low control currents. Whenever required, the following average deratings may be considered... 

In such conditions, it is recommended that the T-R be equipped with an electrical control circuit, which primarily keeps the potential constant, and, in exceptional circumstances, also the protection current. These pieces of equipment are potentiostats (for controlling potential) and galvanostats (for controlling current) [8]. 

Measuring electrodes for impressed current protection are robust reference electrodes (see Section 3.2 and Table 3-1) which are permanently exposed to seawater and remain unpolarized when a small control current is taken. The otherwise usual silver-silver chloride and calomel reference electrodes are used only for checking (see Section 16.7). All reference electrodes with electrolytes and diaphragms are unsuitable as long-term electrodes for potential-controlled rectifiers. Only metal-medium electrodes which have a sufficiently constant potential can be considered as measuring electrodes. The silver-silver chloride electrode has a potential that depends on the chloride content of the water [see Eq. (2-29)]. This potential deviation can usually be tolerated [3]. The most reliable electrodes are those of pure zinc [3]. They have a constant rest potential, are slightly polarizable and in case of film formation can be regenerated by an anodic current pulse. They last at least 5 years. 

The production of metal vapor upon arc interruption necessary to control current chopping and permit voltage recovery and... 

An increase in pressure will also affect the rate of the diffusion of molecules to and from the electrode surface it will cause an increase in the viscosity of the medium and hence a decrease in diffusion controlled currents. The consequences of increased pressure on the electrode double layer and for the adsorption of molecules at the electrode surface are unclear and must await investigation. 

Figure 7. Slow inactivation of Na channels is potentiated by STX. The graph shows the time required for the recovery of Na channels to an activatable state after a long (1 sec, +50 mV) inactivating depolarization. When tested by a brief test pulse, control currents (A) recovered in a fast (r = 233 msec) phase. Addition of STX (q, 2 nM, which approximately halved the currents with no inactivating pulse) approximately doubled the fraction of currents recovering in the slow phase and also increased the time constant of slow recovery. The fast recovery rate was unaffected. (Reproduced with permission from Ref. 47. Copyright 1986 The New York Academy of Sciences).

Before the measurement of HOR activity, a pretreatment of the alloy electrode was carried out by potential sweeps (10 V s ) of 10 cycles between 0.05 and 1.20 V in N2-purged 0.1 M HCIO4. The cyclic voltammograms (CVs) at all the alloys resembled that of pure Pt. As described below, these alloy electrodes were electrochemically stabilized by the pretreatment. Hydrodynamic voltammograms for the HOR were then recorded in the potential range from 0 to 0.20 V with a sweep rate of 10 mV s in 0.1 M HCIO4 saturated with pure H2 or 100 ppm CO/H2 at room temperature. The kinetically controlled current 4 for the HOR at 0.02 V was determined from Levich-Koutecky plots [Bard and Faulkner, 1994]. 

Figure 10.1 Time courses of kineticaUy controlled currents 4 for the HOR at 0.02 V and 26 °C on various electrodes in 0.1M HCIO4 saturated with 100 ppm CO (H2 balance). CO was...

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