Phase angle variation

This behaviour, depicted in Fig. 5.11 with the values calculated using Equation 5.3, represents the phase-angle variations exhibited by Ru/Ti electrodes containing 20-40 at.% Ru, which show low Rt values. 

When a reaction is reversible, i ct—>0 and Zf = Zw = oco 1/2( 1-i). The observed phase angle is jt/4, and the impedance is the least possible for that value of co (o depends not only on the transport but also on the reciprocal of the concentration of electroactive species see (11.19)). If Rct > 0, Zf increases from its minimum value and there will be a lower phase angle. It is this phase-angle variation according to the rate of the electrode reaction that is used in the technique of a.c. voltammetry. 

Figure 24.11 Variation of Trms (left axis) values measured at port 1 x/d = 4) and CO concentration (right axis) of gases sampled from port 4 x/d= 14) with relative phase angle between primary and secondary air driving in the 50-kilowatt forced combustor at China Lake...

The phase angle for maximum EPR intensity of the triplet-state at a modulation frequency of 16 Hz and at 4 K, was observed to change upon the addition of carotenoids II or III to a sample containing TPP or carotenoid IV to a sample containing TPPS. According to Equations 1 and 2 the variation in phase angle (14), 0, indicates... 

The impedance behavior of electrode reactions is often complex but can be conveniently simulated by computer calculations, especially in the case of the method based on kinetic equations (108, 113). The forms of the frequency response represented in terms of the Z versus Z" complex-plane plots and by relations of Z or phase angle to frequency ai or log (o (Bode plots) are often characteristic of the reaction mechanism and involvement of one or more adsorbed intermediates, and they thus provide diagnostic bases for mechanism determination complementary to those based on dc, steady-state rate versus potential responses. The variations of Z versus Z" plots with dc -level potential, in controlled-potential experiments, also give rise to useful diagnostic information related to the dc Tafel behavior. 

FIGURE 1.11 Bode plots representing (a) total impedance versus frequency and (b) phase angle versus frequency variations for a Randles circuit. 

For example, if we have another heavy atom isomorphous derivative available with heavy atom sites different from those found in the first derivative, when the preceding process is repeated, we will get two solutions, one true and one false for each reflection from the second derivative as well. The true solutions should be consistent between the two derivatives while the false solution should show a random variation. Thus, by comparing the solutions obtained from these two calculations, one (the computer) can establish which solution represents the true phase angle. This is the principle of the MIR method. One can also utilize the anomalous scattering (AS) data of the first derivative to resolve the phase ambiguity. In this case, the technique is called the SIRAS approach. If two derivatives and anomalous data are used, then it is called the MIRAS approach. 

In addition to comparing the sum of squares, the experimental and simulated data should be compared by using complex plane and Bode plots. The phase-angle Bode plot is particularly sensitive in detecting time constants. Boukamp proposed to study the residual sum of squares after subtracting the assumed model values from the total impedance data. If the model is valid, the residuals should behave randomly. If they display regular tendencies, it may mean that the model is not correct and further elements should be added. However, the variations of the residuals should be statistically important. 

Figure 2. A vector diagram of two PACD experiments (one of each sign) which illustrates how the PACD experiment may be treated as a superposition of two individual PAS experiments—one with left and the other with right circularly polarized light—that are performed 180° out-of-phase with one another. Here, i and 0r are the PAS phase angles of the left (q,) and right (q,) circularly polarized PAS experiments. The vector sum of q, and qr yields Aq, the vector magnitude of the PACD pressure variations, which has phase c (19).

---