Logarithmic analysis

The transformation by log double-centering has received various names among which spectral mapping [13], logarithmic analysis [14], saturated RC association model [15], log-bilinear model [16] and spectral map analysis or SMA for short [17]. 

According to Andersen [12] early applications of LLM are attributed to the Danish sociologist Rasch in 1963 and to Andersen himself. Later on, the approach has been described under many different names, such as spectral map analysis [13,14] in studies of drug specificity, as logarithmic analysis in the French statistical literature [15] and as the saturated RC association model [16]. The term log-bilinear model has been used by Escoufier and Junca [ 17]. In Chapter 31 on the analysis of measurement tables we have described the method under the name of log double-centred principal components analysis. 

A negative current wave appeared in the polarogram as shown in curve 1 of Fig. 5, though the wave was not observed in the absence of NADH in W or CQ in DCE. The logarithmic analysis of the current wave based on the theoretical equation for the electron transfer [42,54,55] indicated that the wave was caused by two-electron transfer at the interface and controlled by the diffusion of NADH in W. 

FIGURE 1.10. Rotating disk electrode voltammetry. A + e B, with a concentration of A equal to C° and no B in the solution a Linearized concentration profiles —, at the plateau (vertical arrow in b), , at a less negative potential (horizontal arrow in b). b Current potential curve, c Concentrations of A and B at the electrode surface, d Logarithmic analysis of the current potential curve. 

FIGURE 1.11. Convolution of the cyclic voltammetric current with the function I j Jnt, characteristic of transient linear and semi-infinite diffusion. Application to the correction of ohmic drop, a —, Nernstian voltammogram distorted by ohmic drop , ideal Nernstian voltammogram. b Convoluted current vs. the applied potential, E. c Correction of the potential scale, d Logarithmic analysis. 

Figure 6.10 Logarithmic analysis of a polarographic wave by using the Heyrovsky-Ilkovic method (via equation (6.6)). The intercept on the x-axis is an accurate value of Ei/2 for the Fe, Fe couple in water.

From equation (8.4), show why the above logarithmic analysis is suggested. 

Logarithmic analysis of the convolution curves, using the equation holding for irreversible processes, led to very good (r = 0.995—0.999) second-order plots of the potential dependence of log/chei-... 

Convolution potential sweep voltammetry (CPSV) refers to the mathematical transformation of LSV current—potential data resulting in curves with shapes like conventional polarograms which are suitable for logarithmic analysis. The method was first proposed for the study of electrode kinetics by Imbeaux and Saveant [74] but is equivalent in all respects to a semi-integral technique reported earlier by Oldham [75— 77]. A very readable description of the method has been presented by Bard and Faulkner [21]. 

For electrode mechanism analysis, the slope of the appropriate logarithmic analysis of theoretical data is compared with that observed experimentally. CPSV slopes for a number of mechanisms are compared with those from other analyses in Table 25. 

Logarithmic analysis is therefore possible and can be utilised in the determination of E /2 and (an) from quasi-reversible voltammetric waves [159]. For the totally irreversible case, when there is a significant overpotential, we obtain, for all electrodes for a cathodic process... 

At one extreme we will observe two separate one-electron waves and at the other, a two-electron wave [166], The intermediate cases where the waves are overlapping is more difficult to analyse. It is, however, amenable to graphical logarithmic analysis, which has been treated by Rufcic et al. at... 

The slope of this straight line is 16.91 x n V-1 at 25 °C. However, it is more common to use the inverse function E = Ei/2 + 2.303 x (RT/nF) log [(fi, - I) /I], with the slope 0.059/nV. Both functions are called the logarithmic analysis of DC polarogram. They both cross the potential axis at the half-wave potential, which corresponds to I = Ii/2. The main characteristic of fast and reversible electrode reactions is that the half-wave potential is independent of the drop life-time in DC polarography, or the rotation rate of the rotating disk electrode, or the radius of microelectrode. If this condition is satisfied, the slope of the logarithmic analysis indicates the number of electrons in the electrode reaction. 

The current-potential relationship of the totally - irreversible electrode reaction Ox + ne - Red in the techniques mentioned above is I = IiKexp(-af)/ (1+ Kexp(-asteady-state voltammetry, a. is a - transfer coefficient, ks is -> standard rate constant, t is a drop life-time, S is a -> diffusion layer thickness, and

straight line E = Eff + 2.303 x (RT/anF) logzc + 2.303 x (RT/anF) log [(fi, - I) /I -The slope of this line is 0.059/a V. It can be used for the determination of transfer coefficients, if the number of electrons is known. The half-wave potential depends on the drop life-time, or the rotation rate, or the microelectrode radius, and this relationship can be used for the determination of the standard rate constant, if the formal potential is known. 

In the general case, the logarithmic analysis of the wave is a curve with two asymptotes. For the reduction, the slope of the asymptote at higher potential is 0.059/n V, while the slope of the asymptote at lower potentials is 0.059/an V. The half-wave potential depends on the drop life-time, or the diffusion layer thickness. The electrode reactions with these characteristics are called - quasireversible. 

The logarithmic analysis based on the theoretical equation for a reversible electron transfer at the W/0 interface [see Equation (20)] supported that the current was caused by 1 electron transfer at the W/NB interface. 

Hence, for a reversible system, the well-known linear relation is obtained between the potential E and log (/iim -///). Other equations have been derived for those reversible systems that involve semiquinone formation, dimerization, or the formation of complex compounds with mercury. Logarithmic analysis of the polarographic wave is often the only proof of reversibility which is considered but recently several authors, in particular Zuman and Delahay, " have pointed out that it is inadequate to assume that an electrode process is reversible on this evidence alone. For a reversible reaction, plots of E vs. In (/lim - ///) give the electron number z from the slope of the plot, RT/zF, A clearer indication of irreversibility is the evaluation of slopes of log i-E curves for higher concentrations (for i < /lim). Irreversible processes will give Tafel behavior. 

The number of electrons transferred in an electrode process, the shape of the waves (the transfer coefficient a from logarithmic analysis and from d i/j/d pH) and the type and mechanism of the electrode process (based on... 

Palyi et al. (1962) reported that the first wave produced by rhodamine B is a diffusion-controlled wave, while the second wave is an adsorptive wave. Various experiments were performed by Jiao et al. (1982) to verify that Pj is adsorptive in nature. The logarithmic analysis of i-t curves showed that a = 0.24 and 0.55 for P and respectively. The greater value of a means that the corresponding wave is adsorptive. pH exerts a great influence on Pj. At pH 4.5-5.0, the addition of Y(III) causes a decrease of Pj and makes the peak even sharper. This means that the adsorption of the Y(III)-rhodamine-DPG complex is stronger than that of rhodamine itself. 

---