Maxima of the first kind

We saw above that the polarographic current rises from zero to a current plateau. The plateau may be horizontal, or it might be gently sloping upwards we called this rise a residual current. Occasionally, there is also a current peak superimposed on the wave (see Figure 6.32). Such peaks are of two types, i.e. maxima of the first kind and maxima of the second kind. Both are caused by enhanced rates of mass transport at the Hg solution interface, as described in the following. 

A current maximum of the first kind has the form of a sharp, straight line which starts to form just before the main polarographic wave (curve a in Figure 6.32). Such a maximum can be considerably larger than the wave itself, although it will usually drop suddenly back to the normal wave. Maxima of the first kind are caused by convective effects, as electrolyte flows past the surface of the mercury drop, resulting from surface tension differences at various points on the surface of the drop. 

Maxima are caused by streaming of the solution around the surface of the drop . Maxima of the first kind are associated with variations in charge at different places on the drop that occur owing to differences in current density at various positions. Cur-... 

Maxima of the first kind. These appear as sharp straight line enhancements in the current at the beginning of the wave. They can be larger than the wave itself and drop abruptly back to the wave maximum. They are due to convective streaming of the electrolyte past the drop surface caused by differences in surface tension at different points of the drop surface. 

This phenomenon has been briefly discussed in 1.3.3. Sometimes polarographic waves have large maxima. There are two kinds. Maxima of the first kind are found as continuations of the rising part of the wave and are usually sharp (Fig. 1.3f). Maxima of the second kind are rounded humps found on the upper plateau of the wave. Both are adsorption phenomena. Maxima of the first kind are believed to be caused by rapid streaming of the solution past the electrode, somehow caused by adsorbed forms. Maxima of the second kind involves streaming of the mercury within the mercury drop. 

The streaming of solution past the electrode will increase the amount of fresh analyte reaching the electrode and so the current will increase. In maxima of the first kind this only occurs on the rising part of the wave and stops soon after and so the current collapses back to the normal limiting value, thereby forming the sharp maxima. Exactly why adsorption effects should produce this streaming is not clear. 

The maxima of the first kind, arising usually at the beginning of the polarographic wave, are sharp or rounded, and are often observed with low concentrations of electrolytes. Their height varies with the height of the mercury column in different wa, according to the concentration of the supporting electrolyte and the nature of the depolarizer. 

Figure 632 Illustration of polarographic current maxima, where the continuous line is the correct, undistorted polarograph (drawn as the mean current, i.e. without the sawtoothed effect of drop replenishment), and the dashed and dotted lines represent current maxima of the first and second kinds, respectively.

Some Bessel functions of the first kind are shown in Fig. A.1 to illustrate their behavior. The first five zeros of J0 are 2.4048, 5.5201, 8.6537, 11.7915, and 14.9309. The interval between the last two is 3.1394, a value near n. Note that as x increases, the absolute value of the maxima and minima decrease. The larger roots are approximately (m - 1/4)7t, where m is the number of the root. For n > 1/2, the roots approach n from above instead of from below. The first positive zero of J is greater than n, and increase steadily with n. The first zeros are 2.405, 3.832, 5.136, 6.380, 7.588, and 8.771, for n = 0 to 5. The zeros must be found by calculation. 

Besides the mentioned basic currents on polarographic curves, current maxima can occur, according to experimental conditions either of the first kind (tapering shape) or of the second kind (rounded shape). Classical polarography... 

For galvanostatic anodization a first potential maximum is again observed at about 19 V, and the thickness of the anodic oxide at this maxima has been determined to be about 11 nm, as shown in Fig. 5.4. Note that these values correspond to an electric field strength of about 17 MV cm4. The first maximum may be followed by several more, as shown in Fig. 5.1c and d. Note that these pronounced maxima become smeared out or even disappear for an increase in anodization current density (Fig. 5.Id), a reduction in temperature (Fig. 5.1c), or an increase in electrolyte resistivity. The latter value is usually too large for organic electrolytes to observe any current maxima. A dependence of these maxima on crystal orientation [Le4] or doping kind and density [Pa9] is not observed. The rich structure of the anodization curves is interpreted as transition of the oxide morphology and is discussed in detail in the next section. 

The electron density is a continuous function that is experimentally observable, hence uniquely defined, at all points in space. Its topology can be described in terms of the distribution of its critical points, i.e. the points at which the electron density has a zero gradient in all directions. There are four kinds of critical point which include maxima (A) usually found near the centres of atoms, and minima (D) found in the cavities or cages that lie between the atoms. In addition there are two types of saddle point. The first (B) represents a saddle point that is a maximum in two directions and a minimum in the third, the second (C) represents a saddle point that is a minimum in two direction and a maximum in the third. One can draw lines of steepest descent connecting the maxima (A) to the minima (D), lines whose direction indicates the direction in which the electron density falls off most rapidly. Of the infinite number of lines of steepest descent that can be drawn there exists a unique set that has the property that, in passing from the maximum to the minimum, each line passes successively through a B and a C critical point. This set forms a network whose nodes are the critical points and whose links are the lines of steepest descent connecting them. 

HCl in liquid xenon at T 185° K. has been studied (29). A spectrum consisting of three broad maxima is observed in the region where HCl vapor absorbs. The maxima have been interpreted as arising from a vibrational transition where the rotational angular momentum can increase by one quantum, decrease by one quantum, or remain the same. In the gas phase only the first two kinds of transitions are allowed. The third type of transition is expected when an unsymmetrical electrostatic field, such as that existing locally in the liquid, is present. In this case the major contribution to the field arises indirectly from the HCl dipole which induces a dipole in the neighboring polarizable xenon. This reaction field can act back on the HCl dipole. The field is unsymmetrical because of the defect structure of the liquid state. When there is high symmetry, as in the case of HCl in solid xenon (2S), the perturbed spectrum does not occur ... 

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