Bounded Warburg

The bounded Warburg element (BW) describes linear diffusion in a homogeneous layer with finite thickness. Its impedance is written as... 

The Warburg impedance is only valid if the diffusion layer has an infinite thickness. If the diffusion layer is bounded, the impedance at lower frequencies no longer obeys Equation 4.32. Instead, the bounded Warburg element (BW) should be used to replace the Warburg. The impedance of the series connection between the resistance and the BW, shown in Figure 4.9a, can be calculated by adding their impedances ... 

The Nyquist plot is presented in Figure 4.9b. At high frequencies (real axis at a value of R. At low frequencies ( 0), it intercepts the real axis at a value of R+R0. Note that the bounded Warburg impedance is easily recognized from its Nyquist plot. At high frequencies, this circuit element looks like a traditional Warburg impedance and shows a 45° line on the Nyquist plot. At low frequencies, it looks like the semicircle of a Randles cell,... 

Figure D.20. A resistor and a bounded Warburg in series (Figure 4.9a)...

Figure D.21. Nyquist plot of a resistor and a bounded Warburg in series over the frequency range 1 MHz to 1 mHz (R0 = 150 2,

Figure D.56. Modified Randles cell with a bounded Warburg in series with Rct (Figure 4.18a)...

Finite Warburg—for a cathode with fixed diffusion layer thickness (porous bounded Warburg)... 

Riboflavin was first isolated from whey in 1879 by Blyth, and the structure was determined by Kuhn and coworkers in 1933. For the structure determination, this group isolated 30 mg of pure riboflavin from the whites of about 10,000 eggs. The discovery of the actions of riboflavin in biological systems arose from the work of Otto Warburg in Germany and Hugo Theorell in Sweden, both of whom identified yellow substances bound to a yeast enzyme involved in the oxidation of pyridine nucleotides. Theorell showed that riboflavin 5 -phosphate was the source of the yellow color in this old yellow enzyme. By 1938, Warburg had identified FAD, the second common form of riboflavin, as the coenzyme in D-amino acid oxidase, another yellow protein. Riboflavin deficiencies are not at all common. Humans require only about 2 mg per day, and the vitamin is prevalent in many foods. This vitamin... 

Why do we need vitamins Early clues came in 1935 when nicotinamide was found in NAD+ by H. von Euler and associates and in NADP+ by Warburg and Christian. Two years later, K. Lohman and P. Schuster isolated pure cocarboxylase, a dialyz-able material required for decarboxylation of pyruvate by an enzyme from yeast. It was shown to be thiamin diphosphate (Fig. 15-3). Most of the water-soluble vitamins are converted into coenzymes or are covalently bound into active sites of enzymes. Some lipid-soluble vitamins have similar functions but others, such as vitamin D and some metabolites of vitamin A, act more like hormones, binding to receptors that control gene expression or other aspects of metabolism. 

If there are deviations from the ideal semi-infinite linear diffusion process, the bounded Randles cell can also be modified by replacing the Warburg impedance with a CPE. The structure of the model is shown in Figure 4.20a. This modification is applied when the transport limitations appear in a layer of finite thickness. 

W.H. Koppenol, PL. Bounds, and C.V. Dang, Otto Warburgs contributions to current concepts of cancer metabolism, Nat Rev Cancer, 11 (5), 325-37, 2011. 

Our understanding of the extremely important reactions of oxidative phosphorylation in animal tissues has been severely handicapped by technical difficulties, which have as yet by no means been overcome. These difficulties arise from the fact that the enzymes exist in animal tissues in insoluble, particulate form. In 1913, Warburg found that the ability of macerated liver tissue to take up oxygen was found largely in Koernchen or insoluble particles which were readily removed from extracts by centrifugation or by filtration. Many other studies on enzymes concerned with terminal respiration have indicated their insoluble or structure-bound character. Particular attention has been devoted to studies of succinoxidase and cytochrome oxidase. Indeed, these enzymes have come to be considered examples par excellence of insoluble enzymes. 

The boundary conditions for the Warburg impedance, Zw, previously discussed were such that semi-infinite diffusion prevails. However, as we have already seen in connection with voltammetry and other techniques for film-modified electrodes, diffusion in these cases is bounded and is restricted to a thin layer of thickness d. This problem has been independently addressed by three different groups [110-113] and leads to essentially the same end result, namely that the phase angle begins to increase at very low frequencies due to the onset of finite length effects. Figure 20.27a illustrates the complex plane impedance plot obtained in this instance. 

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