Derivative Curves

The appropriate figure for stress, modulus etc. is taken from the creep curve or a derivative curve. This is then inserted into the formula. 

A number of commercial titrators are available in which the electrical measuring unit is coupled to a chart recorder to produce directly a titration curve, and by linking the delivery of titrant from the burette to the movement of the recorder chart, an auto-titrator is produced. It is possible to stop the delivery of the titrant when the indicator electrode attains the potential corresponding to the equivalence point of the particular titration this is a feature of some importance when a number of repetitive titrations have to be performed. Many such instruments are controlled by a microprocessor so that the whole titration procedure is, to a large extent, automated. In addition to the normal titration curve, such instruments will also plot the first-derivative curve (AE/AV), the second-derivative curve (A2 E/AV2), and will provide a Gran s plot (Section 15.18). 

In Fig. 15.7 are presented (a) the part of the experimental titration curve in the vicinity of the equivalence point (b) the first derivative curve, i.e. the slope of the titration curve as a function of V (the equivalence point is indicated by the maximum, which corresponds to the inflexion in the titration curve) and (c) the second derivative curve, i.e. the slope of curve (b) as a function of V (the second derivative becomes zero at the inflexion point and provides a more exact measurement of the equivalence point). 

As has been indicated, if suitable automatic titrators are used, then the derivative curve may be plotted directly and there is no need to undertake the calculations described above. 

A typical conventional polarogram for 0.003 M-cadmium sulphate in 1M potassium chloride in the presence of 0.001 per cent gelatin, and the corresponding derivative curve, are shown in Fig. 16.7 (/max is the maximum current recorded in the derivative mode). 

As shown in Section 15.17, the location of the end point of a potentiometric titration can often be accomplished more exactly from the first or second derivative of the titration curve, than from the titration curve itself. Similarly, absorption observations will often yield more information from derivative plots than from the original absorption curve. This technique was used as long ago as 1955, but with the development of microcomputers which permit rapid generation of derivative curves, the method has acquired great impetus.9,10... 

The influence of an impurity (Y) on the absorption spectrum of a substance (X) can often be eliminated by considering derivative curves as shown in Fig. 17.15 the second-order plot of the mixture is identical with that of pure X. When the interference spectrum can be described by an nth-order polynomial, the interference is eliminated in the (n+ 1) derivative. 

For quantitative measurements peak heights (expressed in mm) are usually measured of the long-wave peak satellite of either the second- or fourth-order derivative curves, or for the short-wave peak satellite of the same curves. This is illustrated in Fig. 17.16(a) for a second-order derivative DL is the long-wave peak height and Ds the short-wave peak height. Some workers11 have preferred to use the peak tangent baseline (DB) or the derivative peak zero (Dz) measurements [Fig. 17.16(h)]. 

For this determination a spectrophotometer which is equipped to produce derivative curves is essential. 

So, corresponding to id for a reversible electrode reaction, Jp is a linear function of concentration the greater sensitivity of the latter permits determinations down to 10 1M(instead of 10 6 Mfor id). Apart from this advantage, the second derivative curve, by means of the difference between its maximum and minimum as a function of concentration, offers an even better check on reaction reversibility32 than the straight-line plot of E (according to eqn. 3.49) against log(tcd - i)/i (see also p. 120), especially because Ip, as a property at the halfwave potential, is more sensitive to the occurrence of irreversibility (cf., pp. 124-127). 

Fig. 3.4037 illustrates well the character of the curves and the gain in sensitivity on going from conventional DC via sampled DC and normal pulse to differential pulse polarography. It should be realized that, although the DPP curve AijAE) might seem an approximation of a derivative curve, we cannot speak of derivative polarography. 

Figure 1.6 First-derivative curves show better apparent resolution than do absorption curves - and second-derivatives curves are still better.

Figure 5.4 First-derivative curves, computed using eqn (5.18) for various values of x, and (oA = coq - 5, T2A-1 = T2B-i = 0.5, pA =Pb = 0-5 note that the vertical scale differs - the plots are magnified by the factors shown.

Perhaps the greatest source of error is introduced by the double integration of the experimental derivative curve. The exact location of the baseline is critical since the outer regions or wings of the spectrum are weighted more heavily than the central portion. One necessary requirement is that the areas enclosed by the curve above and below the baseline must be equal. After the baseline and the initial and terminal points on the spectrum have been determined, the integration can be carried out rather easily by numerical techniques. 

As indicated earlier, one is usually concerned with the derivative curve rather than the absorption curve. The transformation can be made easily by taking the derivative of Eq. (35) which yields... 

This means that the absorption curves in Fig. 12 are replaced by derivative curves. 

The explicit form of the function f (Hr — H, AH) depends on the shape of the individual derivative curves. If the absorption curve can be described by a Lorentzian function, then... 

Here, H is the magnetic field variable and AH is the magnetic field separation between the maximum and the minimum in the derivative curve. This linewidth may be a function of orientation also however, in most calculations it is assumed to be constant. 

Hyperfine interactions likewise produce characteristic inflections in the derivative curve. Anisotropic hyperfine coupling is usually accompanied by anisotropic g values and as a result, the powder spectra are often quite complex. Typical powder spectra for paramagnetic species having one nucleus with / = are shown in Fig. 16. An unambiguous analysis of the more complex experimental spectra often requires the use of two microwave frequencies and a variation in the nuclear isotopes. The latter technique is illustrated by a comparison of the spectra for 14N02 and 15N02 on MgO as shown in Fig. 17. 

Figure 55-6 Expansions of the first and second derivative curves. Figure 55-6a The region around the zero-crossing of the first derivative can be approximated with a straight line. Figure 55-6b The region around the peak of the second derivative can be approximated with a parabola.

Fig. 12. Derivative curves of EPR in a highly dislocated As-doped germanium crystal grown in a H2 atmosphere. The magnetic field is oriented along the [100] direction. T= 2 K, /= 25.16 GHz. Note the sign reversal of the new lines as compared to the As-donor hyperfine structure. Dislocation density 2 x 104 cm 2. (Courtesy Pakulis and Jeffries, reprinted with permission from the American Physical Society, Pakulis, E.J., Jeffries, C D. Phys. Rev. Lett. (1981). 47, 1859.)...

Potentiometric titration curves, (a) Normal curve, (b) First derivative curve. 

Sample and reference crucibles with separate heaters. Thermocouples with feedback to sample heater so that the power is varied to maintain AT= 0. Data output equipment to provide AE vs temperature curves, derivative curves and peak integration. Facility to vary atmosphere of sample. 

Ozawa, T., "Kinetic Analysis of Derivative Curves in Thermal Analysis," /. of Therm. Anal, 2,301 (1979). 

---