Second 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). 

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). 

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

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.

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

Figure 4.5 Plot of A emf against volume for the redox reaction represented by equation (4.1), i.e. the second-derivative curve of the plot shown in Figure 4.2 (as well as being the first-derivative curve of that shown in Figure 4.4).

This is why a dashed line has been drawn near the peak in Figure 4.4. For this reason, a second-derivative curve is often produced, in which case the end point is given by the point of inflection (see Figure 4.5). For a 1 1 reaction (such as that represented by equation (4.1)), the inflection occurs at A emf/AV = 0, and the end point can then be taken as the volume at which the line crosses the axis. 

Using inverse linear interpolation the two titration equivalence points are obtained as the zero-crossing points of the second derivative at V = 3.78 ml and V = 7.14 ml. On Fig. 4.4 the second derivative curve of the interpolating spline (SD = ) and that of the smoothing spline (SD = 8.25) are shown. The false zero-crossing of the second derivative present at interpolation is eliminated by smoothing. 

Figure 11-7 Enlargement of the end-point regions in the second derivative curve shown in Figure 1 l-6c. 

Different experimental approaches are possible with the same endpoint detection method. For example, the titration curve can be plotted and the endpoint determined graphically. First and second derivative curves can be plotted or the derivatives obtained electronically. Another approach is to titrate to a predetermined endpoint signal. This technique is very useful with coulometric titrations, and many examples, especially those involving potentiometric endpoint detection, are found in the literature. The most widely applicable way... 

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. 

Figure 6.6. Potentiometric titration curves, (a) Normal curve, (b) First derivative curve, (c) Second derivative curve...

Fig.2. Second derivative curves from the observed spectra of adsorbed acetonitrile in the 2320-2220 cm range at (a) = 0.05. (b) 0.15 and (c) 0.20.

T Spreadsheet Summary In the final three exercises in Chapter 7 of —I Applications of Microsoft Excel in Analytical Chemistry, we first use Excel to plot a simple distribution of species diagram (a plot) for a weak acid. Then, the first and second derivatives of the titration curve are plotted to better determine the titration end point. A combination plot is produced that simultaneously displays the pH versus volume curve and the second-derivative curve. Finally, a Gran plot is explored for locating the end point by a linear regression procedure. 

Derivative titration curve A plot of the change in the quantity measured per unit volume against the volume of titrant added a derivative curve displays a maximum where there is a point of inflection in a conventional titration curve. See also. second derivative curve. 

Second derivative curve A plot of A-f/AV- for a potentiomet-ric titration the function undergoes a change in sign at the inflection point in a conventional titration curve. 

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