Shape of the Curve

The actual shape of the curve illustrated in Fig. 12.4 can vary according to factors such as metal properties and cavitation intensity. (Cavitation intensity relates to the number of bubbles created in a unit volume of fluid and the amount of energy transferred during the col-... 

The shape of the curves is of some interest. An induction period is noted during which no oxygen uptake, i.e. oxidation, is observed. At the end of this induction period, when in effect antioxidant moieties have been consumed, oxygen uptake is rapid unless some retardation mechanism is at work. 

Complete understanding of the shapes of the curves in Fig. 8.4 requires a kinetic expression somewhat more complicated than we wish to deal with here. The nature of the extremities of the curves can be understood, however, on the basis of qualitative arguments. The rate decreases with a decrease in pH in the acidic region because formation of the zwitterionic tetrahedral intermediate is required for expulsion of the... 

There clearly must be effects from other elements present in the steel incidentally or functionally, but a reasonable relationship between this factor and the measured pitting potential is shown for a large number of steels in Fig. 3.19. Of interest, but not yet explained, is the shape of the curve as discussed elsewhere . 

Anodic E-i curves for nickel obtained by potentiostatic, potentiokinetic or, in earlier days, galvanostatic techniques, have been published by many workers. Unfortunately, good agreement is not always found between data from different sources. The principal reasons for the discrepancies appear to lie in the nature and amount of impurities in the metaP or in the solution -both of which may have a profound effect on the shape of the curve, and in variations in experimental procedure" . 

If the circuit is broken after the e.m.f. has been applied, it will be observed that the reading on the voltmeter is at first fairly steady, and then decreases, more or less rapidly, to zero. The cell is now clearly behaving as a source of current, and is said to exert a back or counter or polarisation e.m.f., since the latter acts in a direction opposite to that of the applied e.m.f. This back e.m.f. arises from the accumulation of oxygen and hydrogen at the anode and cathode respectively two gas electrodes are consequently formed, and the potential difference between them opposes the applied e.m.f. When the primary current from the battery is shut off, the cell produces a moderately steady current until the gases at the electrodes are either used up or have diffused away the voltage then falls to zero. This back e.m.f. is present even when the current from the battery passes through the cell and accounts for the shape of the curve in Fig. 12.1. 

Weak acid with a strong base. In the titration of a weak acid with a strong base, the shape of the curve will depend upon the concentration and the dissociation constant Ka of the acid. Thus in the neutralisation of acetic acid (Ka— 1.8 x 10-5) with sodium hydroxide solution, the salt (sodium acetate) which is formed during the first part of the titration tends to repress the ionisation of the acetic acid still present so that its conductance decreases. The rising salt concentration will, however, tend to produce an increase in conductance. In consequence of these opposing influences the titration curves may have minima, the position of which will depend upon the concentration and upon the strength of the weak acid. As the titration proceeds, a somewhat indefinite break will occur at the end point, and the graph will become linear after all the acid has been neutralised. Some curves for acetic acid-sodium hydroxide titrations are shown in Fig. 13.2(h) clearly it is not possible to fix an accurate end point. 

By means of the experimental methods briefly referred to in 9 a large number of specific-heat measurements have been made at very low temperatures. In Fig. 91 we haye the atomic heats of some metals, and of the diamond, represented as functions of the temperature. The peculiar shape of the curves will. be at once apparent. At a more or less low temperature, the atomic heat decreases with extraordinary rapidity, then apparently approaches tangentially the value zero in the vicinity of T = 0. The thin curves represent the atomic heats calculated from the equation ... 

When the thickness of the laminar sub-layer is large compared with the height of the obstructions, the pipe behaves as a smooth pipe (when e < <5 /3). Since the thickness of the laminar sub-layer decreases as the Reynolds number is increased, a surface which is hydrodynamically smooth at low Reynolds numbers may behave as a rough surface at higher values. This explains the shapes of the curves obtained for plotted against Reynolds number (Figure 3.7). The curves, for all but the roughest of pipes, follow the... 

The influence of the solvent on the oxidation of film under conformational relaxation control is illustrated in Fig. 47, which shows chronoamperograms obtained by steps from -2000 to 300 mV vs. SCE at room temperature (25°C) over 50 s in 0.1 M LiC104 solutions of different solvents acetonitrile, acetone, propylene carbonate, (PC), dimethyl sulfoxide (DMSO), and sulfolane. Films were reduced over 120 s in the corresponding background solution. Despite the large differences observed in the relative shape of the curves obtained in different solvents, shifts in the times for the current maxima (/max) are not important. This fact points to a low influence of the solvent on the rate at which confor-... 

Figure 55. Separation of the overall oxidation curve into its two components a relaxation part [according to Eq. [30]] responsible for the initial shape of the curve, and a diffusion part [Eq. [39]], which controls the final shape of the chronocoulogram. (Reprinted from T. F. Otero and H.-J. Grande, Reversible 2D to 3D electrode transition in polypyrrole films. Colloid Surf. A. 134, 85, 1998, Figs. 4-9. Copyright 1998. Reproduced with kind permission of Elsevier Science-NL, Sara Burgerhartstraat 25, 1055 Amsterdam, The Netherlands.)...

Figure 63. Theoretical simulation of voltammograms obtained from -2500 to 300 mV, showing the effect of the temperature on the shape of the curve and the position of the maximum. (Reprinted from T. F. Otero, H.-J. Grande, and J. Rodriguez, J. Phys. Chem. 101, 8525, 1997, Figs. 3-11, 13. Copyright 1997. Reproduced with permission from the American Chemical Society.)...

One of the most problematic issues, still to be fully resolved, is the dependence of the degree of oxidation on potential, as measured most commonly by cyclic voltammetry at low scan rates. There is currently no accepted model to describe the shape of the curve and the hysteresis between anodic and cathodic scans. The debate over whether the charge has a significant component due to a polymer/solution double layer is still not fully resolved. 

FIGURE 9.18 The variation of the extent of saturation of myoglobin (Mb) and hemoglobin (Hbl with the partial pressure of oxygen. The different shapes of the curves account for the different biological functions of the two proteins. 

We have already seen how to estimate the pH of the initial analyte when only weak acid or weak base is present (point A in Fig. 11.6, for instance), as well as the pH at the stoichiometric point (point S). Between these two points lie points corresponding to a mixed solution of some weak acid (or base) and some salt. We can therefore use the techniques described in Toolbox 11.2 and Example 11.6 to account for the shape of the curve. 

Verification of the shape of the curve has been obtained with the collaboration of Dr. J. Sherman by the theoretical treatment of a somewhat similar problem (the effect of s-p hybridization of bond orbitals on interatomic distance). 

An obvious refinement of the simple theory for cobalt and nickel and their alloys can be made which leads to a significant increase in the calculated value of the Curie temperature. The foregoing calculation for nickel, for example, is based upon the assumption that the uncoupled valence electrons spend equal amounts of time on the nickel atoms with / = 1 and the nickel atoms with J = 0. However, the stabilizing interaction of the spins of the valence electrons and the parallel atomic moments would cause an increase in the wave function for the valence electrons in the neighborhood of the atoms with / = 1 and the parallel orientation. This effect also produces a change in the shape of the curve of saturation magnetization as a function of temperature. The details of this refined theory will be published later. 

It is basically irrelevant in steady-state measurements in which direction the polarization curves are recorded that is, whether the potential is moved in the direction of more positive (anodic scan) or more negative (cathodic scan) values. But sometimes the shape of the curves is seen to depend on scan direction that is, the curve recorded in the anodic direction does not coincide with that recorded in the cathodic direction (Eig. 12.3). This is due to changes occurring during the measurements in the properties of the electrode surface (e.g., surface oxidation at anodic potentials) and producing changes in the kinetic parameters. 

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