Dimensionless sweep rate

Table 6 shows the results obtained for the reversible reduction O + ne- R. For an irreversible reduction with soluble product, the three regions of peak behaviour dependent on dimensionless sweep rate, a, are the same as for a reversible reaction. In this case, for a > 10... 

Note the rather simple form of the Nernst equation here, and the fact that the dimensionless sweep rate is now unity, that is, one p-unit per one T-unit. 

We can now define a reference time r, being the time taken for the potential to change by one p-unit. Normalizing time t as above (Eq. 8), that isT = tjx (r = a ), we now have a sweep starting at p and moving, for all experimental parameter sets, with the dimensionless sweep rate 1, as is seen in the new dimensionless... 

Example Determination of ks from AEp Measurements The ks value of a quasireversible system is conveniently determined by monitoring the peak separation of the CV wave as a function of sweep rate, using the results of calculations by Nicholson [12] that relate AEp to the dimensionless parameter vj/, through Equation 23.12 ... 

To render the LSV system dimensionless, the usual reference values for concentration, time and distance from the electrode are needed, as well as that for potential (2.29) (and thus, sweep rate). Both species concentrations are normalised by the initial bulk concentration of A, c, as always, and the potential to dimensionless p as in (2.29), (2.89) thus becoming... 

First we must normalise some quantities, to make them compatible with the other dimensionless parameters already used. We refer to the normalisation formulae on p.26. Recall that we have normalised voltage by the factor and that the time unit r for LSV is equal to (v being the sweep rate), or the time the sweep takes to traverse one normalised potential unit V-... 

The symbol P is normally rendered as p, but this collides with our p for the dimensionless potential. For LSV, a small P value means a slow LSV sweep rate and a sigmoidal steady-state response, while a large value means a fast sweep rate, with the UMDE behaving more like a planar (shrouded) electrode. It will be seen later that the above impinges on the choice of a maximum Z... 

Dimensionless heterogeneous rate constant Chemical potential Standard chemical potential Sweep rate in CV or LSV (Vs )... 

Figure 6.29 shows some example linear sweep voltammograms assuming different scan rates (Osrrefers to the dimensionless scan rate Osr = F/RT)(yrl/D)). As the experimental time scale decreases, the diffusional behavior changes from near-steady-state to near-planar diffusion. With respect to the different shapes of microparticles, the mass transport-limiting current was found to be fairly consistent that is, a difference of less than 2% for sphere and hemispheres of equal surface area. 

Fig. 45. Linear sweep voltammogram of a reversible process in terms of dimensionless current function according to Eq. (79). The sweep rate parameter a is defined in Eq. (77).

Notes LSV linear sweep voltammetry. CV cyclic voltammetry, DCP direct current polarogra-phy, MEV steady-state voltammetry at microelectrodes, ChP chronopotentiometry, RDE rotating disc electrode, TEV voltammetry in tubular electrodes, kf and k, rate constants of irreversible follow-up and catalytic reactions, respectively cj, the bulk concentration of the catalyst Z k = kf + k, sum of the rate constants and drop time and transition time, respectively a radius of a hemispherical microelectrode, Xf length of the tubular electrode in the flow direction, co the angular rotation rate, Uf linear flow rate in a TE (7 = RT/(nFv) is the time required for the dimensionless potential shift nFdE/RT at the sweep rate v (Vs" ). 

Ip d- Another form of normalization is used in the CV method, in which the ratio of anodic and cathodic peak currents, Ipb/Ipf (respecting the backward and forward scans), is determined. For comparison of experimental and simulated responses working curves are often used, involving normalized data plotted against appropriate dimensionless kinetic parameters. The most important is the kinetic parameter X the formulations of which for six experimental techniques are given in Table 2. Vafues of X are always products of chemical rate constants (of s" dimension) and time (drop life-time, reciprocal sweep rate or reciprocal frequency) in s-scale. The choice and the application of this parameter will be discussed in the next two subsections. 

The kinetic analysis of a complicated electrochemical process involves two crucial steps the validation of the proposed mechanism and the extraction of the kinetic parameter values from experimental data. In cyclic voltammetry, the variable factor, which determines the mass transfer rate, is the potential sweep rate v. Therefore, the kinetic analysis relies on investigation of the dependences of some characteristic features of experimental voltammograms (e.g., peak potentials and currents) on v. Because of the large number of factors affecting the overall process rate (concentrations, diffusion coefficients, rate constants, etc.), such an analysis may be overwhelming unless those factors are combined to form a few dimensionless kinetic parameters. The set of such parameters is specific for every mechanism. Also, the expression of the potential and current as normalized (dimensionless) quantities allows one to generalize the theory in the form of dimensionless working curves valid for different values of kinetic, thermodynamic, and mass transport parameters. 

Choosing a proper potential sweep rate (v) in pipette voltammetry is essential for attaining a steady-state and sufficiently low charging current. Computer simulations and experiments showed that ion diffusion on either side of the nano-ITIES reaches a steady state dnring a potential cycle at a moderate The related dimensionless parameter g= a lAD.,) z,FvlRT) com-... 

This represents a readily available range these limits can often be extended to shorter times under favorable conditions. For example, potential and current steps in the nanosecond range and potential sweeps above 10 V/s have been reported. This time window should be considered only approximate. A better description of the conditions under which a chemical reaction will cause a perturbation of the electrochemical response can be given in terms of the dimensionless rate parameter, A, discussed in Section 12.3. 

The mass transfer coefficient in linear sweep voltammetry and cyclic voltammetry is directly proportional to the square root of the potential scan rate Accordingly, the apparent reversibility of an ET reaction under voltanunetric conditions is determined by the value of the dimensionless parameter A = k° jRT/FDv (4), and the kinetic zones can be specified as follows ... 

For LSV, the diffusion equation for a UMDE is a little different from that for a potential jump. The LSV case can be considered as one of a group of possible cases, in which the characteristic time is defined independently of the disk radius while the space variables are rendered dimensionless using the disk radius, as described before. In general, let that characteristic time be t. For LSV, as described on page 29, it is the time taken by the potential to sweep over one dimensionless potential unit and the scan rate parameter a was introduced, Eq. (2.91). It has the dimension of s so that r = Note that the scan rate parameter a should not be confused with the radius of a UMDE which has the same symbol. The dimensionless time is then defined by... 

Note that P = 1 for the simulation of a potential step experiment, so that Eqs. (12.39) and (12.40) become identical to Eqs. (12.37) and (12.38). In the case of LSV Tmax is the time to scan a potential from a starting value, Estart, taking some potential units RT/nT, to the final potential Estop with a scan rate v. The characteristic time x was previously defined as the time to sweep through one p-unit, see Eq. (2.92) on page 29. Therefore Ty ax is equivalent to the dimensionless potential range Prange of the LSV simulation and Z ax becomes... 

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