First derivative curve

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). 

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.

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

Nitrate (a) Sigmoid (Regular) Curve (b) First Derivative Curve (c) Second Derivative Curve. 

Figure 16.1 (b) is obtained by plotting AE/AV against V which is termed as the first derivative curve. It gives a maximum at the point of inflexion of the titration curve i.e., at the end-point. 

Figure 4.4 Plot of Aemf IAV against volume, i.e. the first-derivative curve of the plot shown in Figure 4.2, for the redox reaction represented by equation (4.1).

Conversely, a better approach involves drawing a plot of Aemf against AV, i.e. obtaining the first-derivative curve (as shown in Figure 4.4), where the end point is now given by a peak. ... 

Exercise 27-13 The esr spectrum shown in Figure 27-18 is a first-derivative curve of the absorption of a radical produced by x irradiation of 1,3,5-cycloheptatriene present as an impurity in crystals of naphthalene. Sketch this spectrum as it would look as an absorption spectrum and show the structure of the radical to which it corresponds. Show how at least one isomeric structure for the radical can be eliminated by the observed character of the spectrum. 

The end point in a potentiometric titration can be determined by one of the following three methods Direct plot, first-derivative curve, and second-derivative curve. 

A first-derivative curve may alternatively be constructed by plotting change in potential per unit volume, i.e., AE/mL of titrant, added against the mL of the titrant. The curve would appear like a sharp peak, as shown in Figure 1.6.5. The end point is the volume corresponding to the highest AE/mL, i.e., the tip of the peak. 

Fig. 35. Characteristic line shape for a) Lorentzian and b) Gaussian absorption curve, together with functions for absorption and first derivative curves...

Figure 38 shows results obtained by Carstensen from UVirradiated ds-l,4-poly-isoprene at 77 K. Only 4 lines were clearly defined in the first derivative curve. 

Figure 38 shows the result of integrating the first derivative curve to obtain... 

Figure 21-21 Titration of 2.433 mmol of chloride ion with 0.1000 M silver nitrate, (a) Titration curve, (b) First-derivative curve, (c) Second-derivative curve.

Using first derivative curves of signal as a function of field strength at a frequency of 100 K Hertz, Mn2+ was positively identified in our work by the 55Mn2+ nuclear moment. Identification of Mn2+ in all cases was made by the characteristic and well-known six-line band of Mn2+ shown in Figure 1. To simplify the comparison of the six-line band, all... 

A Gran-type plot can also be obtained by plotting the reciprocal of a first derivative curve, that is, AEIAV versus V. Since in a derivative titration I VILe goes to infinity at the equivalence point, the reciprocal will go to zero where the intersection of the two fines occurs, and a V-shaped plot results. In this application, the average volume between the two increments is plotted, as in the first derivative plot. The AE/AF values must be corrected for volume changes to obtain straight lines (AE/AF is linearly dependent on volume changes). 

Fig. 28. Absorption and first derivative curves for radicals with a g tensor that is (a) axially symmetric and (b) anisotropic M).

ESR lines usually exhibit shapes very close to those of Gaussian or Lorentzian functions. The intensities of ESR lines may be obtained by integration of the full absorption curve, by two consecutive integrations of the first-derivative curve, or by the approximation... 

Applying first principles to measurements of the ESR signal and to all pertinent instrumental parameters, absolute numbers of spins can be determined. However, this is rarely done since the number of variables to be controlled is considerable and the labor involved is disproportionate. Relative concentrations of species with the same spectral shape and line widths can be determined simply by comparing peak heights of the normal first-derivative curve under identical conditions, i.e.. 

Figure 30. Cooling curve, first derivative curve and representation of characteristic parameters used and analyzed in this evaluation of a 319 aluminium alloy [44]. (With permission fi om Elsevier.)...

In the TMA plot in Fig. 7 it is possible to note the interactive nature of the substrate on the curing of the PF adhesive. For example, the modulus of elasticity (MOE) increase curve shows two sections (and a two peak first derivative curve). This indicates formation of entanglement networks of the resin in wood which is not possible on noninteractive substrates such as glass as in Fig. 6. Of course DMA and TMA give equally good results when used on the same wood substrate [379,380]. The ABES technique is also linearly correlated with TMA and DMA results as has been demonstrated by the linear relationship that has been found for both MUF and tannin formaldehyde adhesives in the results of TMA and ABES [381]. 

Figure 7 TMA plot of the curing of a PF resin on beech wood. Heating rate 10°C/min. Numbers in the figure are temperatures in °C. (After ref. 370.) (O) MOE curve (A) first derivative curve.

TPeak = Temperature of the Decomposition Peak evaluated as the minimum of the first derivative curve, WL DPI, WLP2 = % of Weight loss in respectively the first and the second decomposition peak. Residue = % of Weight Residue at 600 C... 

When submitted to TGA, SCB decomposed in one step with a minimum in the first derivative curve at 342 C, while WG decomposed mainly in two overlaid steps with minimum peaks at 276 C and 305 °C respectively. WGSCB20 thermal stability (Ton = 223 C) was sligthly lower than WG (Ton = 237 °C) and WGSCB20 minimum peak (T= 312 C) occurred between WG and SCB minimum peaks (Fig. 3). 

Figure 10 CPM-DSF characterization of severai cysteine mutant pairs for a CC chemokine and its receptor. (A) Meiting curves piotting the increasing fluorescence of the CPM dye as a function of temperature. (B) First derivative curves of (A) demonstrate the positions of the peaks used to caicuiate T. (Q Meiting temperatures are caicuiated from the derivative curves. One mutant combination stands out as having a significantiy higher T. ...

Fig. 6.14 Potential response to a current step —1.45 mA cm (curve 1) and its first derivative (curve 2)...

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