Mass-transport-controlled currents

In this section, we consider mass transport-controlled currents to disc and concentric ring electrodes on a planar spinning disc surface. For other less common rotating electrodes, e.g. rotating hemisphere, see Table 3. 

Thus, as the resistivity increases, the iR drop can become excessive. This problem is somewhat ameliorated by the fact that diffusion coefficients also decrease with decreasing temperature so that mass-transport-controlled currents will be smaller. However, this effect is not large enough to offset significantly the deterioration in response due to increased resistance. 

It is evident then that the overall rate constant is determined by the lowest rate constant. We note, in passing, that Eq. 12F is similar to Eq. 3A, which correlates the overall current to the activation and mass-transport controlled currents. 

It will be apparent from the previous section that ion-selective electrodes can be made suitable for analysis in the field and that, in circumstances where the overall composition of the medium is not too variable, they may also be used directly to monitor metal ion and/or anions in plant streams, effluents, water supplies and even rivers. In such applications of ion-selective electrodes, however, it is necessary to avoid electrical pick-up and ground loops by correct placement and shielding of the electrodes. There are also many other situations where the capabiUties of electrochemistry and the characteristics of an analysis can be matched to allow the manufacture of on-line or portable devices. The actual measurement may be the potential of an ion-selective electrode, the mass-transport-controlled current at an electrode held at constant potential or even conductivity. This latter measurement, for example, remains the optimum way of estimating the total salt concentration in solution. 

O + ne"--------+R showing a mass transport controlled current (4 as a plateau in the... 

The great usefulness of the rotating disk electrode is that the mass transport controlled current is proportional to the square root of the rotation speed. If the rate of the electrode process is controlled by both the rate of electron transfer and of mass transport, it can be shown that the current flowing through the cell is related to the rotation speed by the following equation... 

So far, we have discussed the properties of the limiting current density at the RDE. What about activation or mixed control For a purely activation-controlled process, the current should be independent of rotation rate, or should at least become independent of it beyond a certain rotation rate. Under conditions of mixed control, the activation-and mass-transport-controlled current densities combine to yield the total current density as the sum of reciprocals, namely... 

In this section we consider experiments in which the current is controlled by the rate of electron transfer (i.e., reactions with sufficiently fast mass transport). The current-potential relationship for such reactions is different from those discussed (above) for mass transport-controlled reactions. 

Use equations to demonstrate how an increase of the stirring rate will effect the mass transport-controlled limiting current. 

Overall, the RDE provides an efficient and reproducible mass transport and hence the analytical measurement can be made with high sensitivity and precision. Such well-defined behavior greatly simplifies the interpretation of the measurement. The convective nature of the electrode results also in very short response tunes. The detection limits can be lowered via periodic changes in the rotation speed and isolation of small mass transport-dependent currents from simultaneously flowing surface-controlled background currents. Sinusoidal or square-wave modulations of the rotation speed are particularly attractive for this task. The rotation-speed dependence of the limiting current (equation 4-5) can also be used for calculating the diffusion coefficient or the surface area. Further details on the RDE can be found in Adam s book (17). 

Sum and Skyllas-Kazacos [44] studied the deposition and dissolution of aluminum in an acidic cryolite melt. The graphite electrode was preconditioned (immersed in cryolite melt) to saturate the surface of the electrode in sodium before aluminum deposition could be observed. Current reversal chronoamperometry was used to measure the rate of aluminum dissolution in the acidic melt. Dissolution rate was mass transport controlled [45] and in the order of 0.8 10 7 and 1.8 10 7 molcm 2s 1 at 1030 °C and 980 °C respectively [44]. 

We will now combine the Levich and Butler-Volmer approaches. The Levich relationship (equation (7.1)) is written in terms of the limiting current / jm, where limiting here means proportional to Canaiyte - in other words, the electrode reaction is so fast that the magnitude of the current is controlled only by the flux of analyte to the electrode solution interface, i.e. /um is mass>transport controlled. 

Instruction Calculate the current density values as a function of overpotential (in a range of -0.200 to 0.200 V) assuming that the reaction is under mass transport control and under mixed mass transport and charge-transfer control determine the error of the approximation and plot i-T) dependencies. (Gokjovic)... 

In the previous section, the velocity and concentration distributions have been established and two transfer functions were considered. The explicit form of the third function which relates the fluctuating interfacial concentration or concentration gradient to the potential or the current at the interface, requires to make clear the kinetic mechanism composed of elementary steps with at least one of them being in part or wholly mass transport controlled. 

Equations (1.183) and (1.184) point to the existence of maximum or limiting currents that can be obtained under mass transport control conditions. These limiting currents correspond to the case of null surface concentrations of the electroactive species, i.e., for Aci = c, ... 

For more negative potentials, the current (solid line) deviates from the activation control (given by the dashed line which corresponds to a pure kinetic behavior9) and begins to be influenced by the mass transport (second term in Eq. (1.193), which in practice means that c / c ), until for certain potentials at which the mass transport controls the overall current (erl —> 0 and kKa —> oo) and under these conditions... 

This equivalence between the charge of surface-bound molecules and the current of solution soluble ones is due to two main reasons first, in an electro-active monolayer the normalized charge is proportional to the difference between the total and reactant surface excesses ((QP/QP) oc (/> — To)), and in electrochemical systems under mass transport control, the voltammetric normalized current is proportional to the difference between the bulk and surface concentrations ((///djC) oc (c 0 — Cq) [49]. Second, a reversible diffusionless system fulfills the conditions (6.107) and (6.110) and the same conditions must be fulfilled by the concentrations cQ and cR when the process takes place under mass transport control (see Eqs. (2.150) and (2.151)) when the diffusion coefficients of both species are equal. 

The study of the variation of the current response with time under potentiostatic control is chronoamperometry. In Section 5.4 the current resulting from a potential step from a value of the potential where there is no electrode reaction to one corresponding to the mass-transport-limited current was calculated for the simple system O + ne-— R, where only O or only R is initially present. This current is the faradaic current, If, since it is due only to a faradaic electrode process (only electron transfer). For a planar electrode it is expressed by the Cottrell equation4... 

These expressions assume mass transport control kinetic limitations complicate the analysis. However, it is reasonable to assume that a potential corresponding to the mass-transport-limited current can almost always be chosen. 

The effect of altering the rate of mass transport to the electrode surface was also studied (see Fig. 2.20). At low rotation rates, the reaction is mass transport-controlled but as the rotation speed is increased, the current tends to a rotation speed-independent value indicating that the current becomes limited by some other process. 

It is assumed that all electrons transfers from the particle conduction band and surface states to the electrode take place under conditions where the current is mass transport controlled. The first order rate constant kg describes electron promotion either by thermal or photonic processes, and the rate constant k describes the loss of the electrons from the conduction band or surface states by a process which is first order in electron concentration. The validity of this assumption will be discussed later. There will be an equation similar to equation (71) for each value of m. If each equation is multiplied by its value of m and the engendered set of equations summed, it is possible to obtain the simple result that ... 

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