Differential curve

Fig. 1.11. The normalized angular momentum correlation function Kj(t)/Kj(0) at k — 0.25 in differential (curve a), integral (curve b) and impact (curve c) theories.

Fig. 3.13. Differential curve of growth for Fe i in p, Cas relative to the Sun, after Catchpole, Pagel and Powell (1967).

In (b) typical trends of temperature vs. time are presented T0 is the controlled furnace temperature trend, 7 R is the T trend of the inert reference specimen, Ts is the trend observed for a sample undergoing some transformation. The corresponding differential curve (AT vs. time (or vs. temperature)) is shown in (c). 

As shown in Figure 42, the square wave voltammogram, as in DPV, is peak-shaped, but it consists of a differential curve between the current recorded in the forward half-cycle and the current recorded in the reverse half-cycle (pay attention to the fact that, since the forward and the reverse currents have opposite signs, their difference corresponds in absolute to their sum). 

The line widths recorded are the distances, in gauss, between the two maxima of the differential curve and thus represented the width of the absorption curve at the level of maximum slope. For a curve of Lorentzian shape, this width is equal to 0.577/T2, where T2 is the spin-spin or transverse relaxation time (12). Since the curves were not strictly Lorentzian (though they were so to well below the region of maximum slope) and in any case were not of identical shape, the recorded line widths do not have this exact theoretical significance, and small differences in width between two samples could be caused by shape differences rather than variations in relaxation phenomena. Nevertheless, since the conclusions drawn later from the observed line widths depend on major differences in level rather than subtle differences in numerical value, they should be perfectly valid. 

The differential curves are peak shaped in all cases even under the stationary state, with a peak potential equal to the formal potential if the current is plotted versus 1Ildex (given by equations (7.3) or (7.7)), and the peak current is given by... 

With the di-sodium salt of 1,2-dicarbomethoxyethylphosphonic acid as modifier the diethylene glycol content and the carboxyl group content is very low (Table III). These results indicate that the used modifier has also thermostabilizing properties. The integral and differential curves (Fig.2) of molecular mass distribution show that the polydispersity of modified resin is comparable to that of the un-modified. 

Figure I. Molecular mass distribution curves from phosphorus-containing poly-ethyleneterephlhalate sample with 0.74% phosphorus modified with sodium salt of diethyl phosphite. Key l, integral curve 2. differential curve and 3, integral curve of phosphorus distribution.

Potentiometric titration curves normally are represented by a plot of the indicator-electrode potential as a function of volume of titrant, as indicated in Fig. 4.2. However, there are some advantages if the data are plotted as the first derivative of the indicator potential with respect to volume of titrant (or even as the second derivative). Such titration curves also are indicated in Figure 4.2, and illustrate that a more definite endpoint indication is provided by both differential curves than by the integrated form of the titration curve. Furthermore, titration by repetitive constant-volume increments allows the endpoint to be determined without a plot of the titration curve the endpoint coincides with the condition when the differential potentiometric response per volume increment is a maximum. Likewise, the endpoint can be determined by using the second derivative the latter has distinct advantages in that there is some indication of the approach of the endpoint as the second derivative approaches a positive maximum just prior to the equivalence point before passing through zero. Such a second-derivative response is particularly attractive for automated titration systems that stop at the equivalence point. 

Parameters of the porous structure of titania samples (pores volume Vs, specific surface area Ssp) were calculated using BET theory [34] from the adsorption isotherms of methanol. The average pore diameter (Dp) values were estimated from the differential curves of pore size distribution. 

Fig. 3.3 Schematic plots of the double layer region, (a) Electrocapillary curve (surface tension, y, vs. potential) (b) Charge density on the electrode, aM, vs. potential (c) Differential capacity, Cd, vs. potential. Curve (b) is obtained by differentiating curve (a), and (c) by differentiation of (b), Ez is the point of zero...

Here it is assumed that y > ft for all values of penetrant concentration. In Fig. 10 is shown the family of successive differential curves calculated from Eq. (1) with Eq. (17) as the boundary equation in these calculations... 

As mentioned above an increase of the linewidth occurs by the superposition of the lateral and central gaussian curves or by a shift of the extremes of the differentiated absorption curves, which is essentially the same. Again the real linewidths can be obtained iteratively only. For this purpose we begin with the twice-differentiated form of Equation 8 because the question after the shift of the extremes for the differentiated curves is identical to the question after the shift of the zeros for the twice-differentiated absorption curves. There the twice-differentiated Equation 8 is developed in a Taylor series at the positions of the extremes breaking off after the second term. At these positions the function is... 

The differential curve (Y ) is obtained by differentiating the normal output,... 

The differential form of the Gaussian function has already been discussed and is sigmoid in shape with a positive maximum at the first point of inflexion of the Gaussian curve and a minimum at the second point of inflexion. If the peaks are completely resolved in the normal chromatogram, then they can be clearly and unambiguously identifiable in their differential form. If, however, the peaks are not completely resolved, then the differential curve of the unresolved peaks are confused and extremely difficult to interpret and for this reason the differential form of the Gaussian function is rarely used. Nonetheless, if the elution profile of the solutes are not Gaussian in form, the differential detector can be extremely useful. 

Fig. 6. Rate of change of solubility with ammonium sulfate concentration, obtained by differentiating curves of Fig. 5.

For a given protein, the distribution curves for different values of Co are all determined by the same differential curve. Co merely determines the point of entry to the curve. Taking the case of two solutions of carboxy-myoglobin, one containing 30 gm per liter and the other being a tenfold dilution of this, we read off from Fig. 5 the respective salt concentrations for first precipitation (points A and B) and transfer them to Fig. 6. The two distribution curves of the precipitation will now be portions A to C and B to C of the differential curve in Fig. 6 for the stronger and weaker solutions respectively. The two peaks represented by the areas AA C and BB C are of course of different sizes because of the different amounts of protein taken, but if they are normalized by expressing the amounts as... 

Fig. 2 a, b. Differential curves of the distribution of cell radii for rigid PUR foam based on polyester (grade PPU-307) of the apparent densities 40 kg m (a) and 500kg m (b). 

The ratio of the sums in the numerator and denominator of Eq. (10) is equal to unity only for the closest sphere packing of monodisperse cells for all other types of packing and for poly-disperse cells this ratio is always less than unity. Differential curves of the specific surface distribution over cell sizes (Curves 2 in Fig. 2a, b) are described by Eq. (11) ... 

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