Titration second-derivative method

The second-derivative method is an extension of the first-derivative method. The second-derivative of the data changes sign at the point of inflection in the titration curve. This change is often used as the analytical signal in automatic titrators. 

Q Spreadsheet Summary In the second experiment in Chapter 11 of Applications of Microsoft Excel in Analytical Chemistry , a spreadsheet is developed to plot a coulometric titration curve. The end point is located by first- and second-derivative methods. 

Another method for finding the end point is to plot the first or second derivative of the titration curve. The slope of a titration curve reaches its maximum value at the inflection point. The first derivative of a titration curve, therefore, shows a separate peak for each end point. The first derivative is approximated as ApH/AV, where ApH is the change in pH between successive additions of titrant. For example, the initial point in the first derivative titration curve for the data in Table 9.5 is... 

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... 

With today s titrimeters the titration can be programmed so that not only the curve is directly registered but also its first derivative and often even its second derivative. Once the empirical curve has been obtained, a method of end-point detection must be applied, and this should be such that the end-point detected agrees with the true equivalence point. 

New methods can be created by automatic optimization of parameters during a trial run and all methods can be stored permanently in a non-volatile area of memory which is preserved even when the instrument is switched off. Some instruments provide a means of producing first and second derivatives of the titration curve (p. 243) which can be advantageous where the end-point is indistinct or there is more than one end-point to be detected. Titrators with a substantial amount of RAM incorporate what is in effect a dedicated microcomputer. 

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. 

Fig. 13.2. Methods for determining the equivalence point of a potentiometric titration curve (including acid-base titrations), (a) First derivative (b) Second derivative (c) Gran plot for titration of a strong acid with a strong base Vx is the initial volume of acid and V the volume of base added.

By differentiating the titration curve twice and then equating the second derivative to zero, it can be shown that for a symmetrical titration curve ( i = the point of maximum slope theoretically coincides with the equivalence point. This conclusion is the basis for potentiometric end-point detection methods. On the other hand, if 2> the titration curve is asymmetrical in the vicinity of the equivalence point, and there is a small titration error if the end point is taken as the inflection point In practice the error from this source is usually insignificant compared with such errors as inexact stoichiometry, slowness of titration reaction, and slowness of attainment of electrode equilibria. 

The location of the end-point of a titration by using either the first or second derivative of the titration data was discussed in Chapter 9. These methods use only the data points near the end-point. Another approach. Gran s method, makes use of the complete data set. It is useful when either (i) the inflection at the end-point is poorly defined or (ii) data at the end-point is missing. 

A second approach is to calculate the change in potential-per-unit change in volume in reagent (AE/AV). By inspection, the endpoint can be located from the inflection point of the titration curve. This is the point that corresponds to the maximum rate of change of cell emf per unit volume of titrant added (usually 0.05 or 0.1 mL). The first-derivative method is based on the sigmoid shaped curve. 

Spreadsheet Summary In Chapter 7 of Applications of Microsoft Excel in Analytical Chemistry, the first and second derivatives of an acid/base titration curve are plotted in order 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, an alternative plotting method, known as a Gran plot, is explored for locating the end point by a linear regression procedure. 

The same authors used the inflection point of titration curves to find the PZC from a charging curve for one ionic strength. The dtro/dpH is plotted versus pH and the maximum indicates the PZC. The point of zero charge corresponds to the inflection point of titration curves (second derivative of uq versus pH = 0). [66]. Sometimes the cTo-pH dependence is linear [67], and in such case the infection point method to find the PZC cannot be applied. An example of application of the inflection point method to authentic experimental data (IniOj and In(OH)3), Hamada et al., cf Table 3.1) is given in Figs. 3,7 and 3.8 (first and second derivative). The match between originally claimed PZC and that obtained from the first and second derivatives of [Pg.83]

FIG. 3.8 Inflection point method (second derivative). The arrows indicate the PZC from titration. Calculated from uncorrected titration curves published by Hamada et al. (1990). 

Analytically useful acid-base titration curves are characterized by a rather fast pH change near the equivalence point. This suggests that the location of the equivalence point might be determined experimentally from that of the maximum in its first derivative, d(pH)/dVfo, or the zero-crossing of its second derivative, d2(pH)/dVj,2. The advantage of such an approach is that it does not rely on any particular theoretical model, but instead exploits the characteristic feature of the titration curve, i.e., its fast pH change in the region around the equivalence point. The method does not even require that the pH meter is carefully calibrated. 

A word of caution should be mentioned with respect to derivative methods. The derivatives tend to emphasize noise or scatter in the data points, being worse for the second derivative. Hence, if a particular titration is subject to noise or potential drift, a direct plot may be preferred. 

There are numerous automatic titrators that employ potentiometric end-point detection. They usually can automatically record the first or second derivative of the titration curve and read out the end-point volume. The sample is placed in the titration vessel, and the titrant, drawn from a reservoir, is placed in a syringe-driven buret. The volume is digitally read from the displacement of the syringe plunger by the electronic driver. Titrators may also employ photometric detection of indicator color changes. An automatic titrator is shown in Figure 14.7. Automatic titrators make volumetric analyses rapid, reproducible, and convenient. While instrumental methods provide many advantages, classical volumetric analyses are still widely used and are very useful, especially for major constituents, for example, in the pharmaceutical industry. 

The performance of the titration can be controlled In a variety of ways (see Table 13.1) by use of empirical equations for the calculation of AV from preceding titration data points by use of microprocessors to control volumetric equipment (e.g. in photometric, potentlometrlc, coulometrlc titrations) or expand the scope of a given technique by use of robot stations In Implementing laborious manual methods or In handling toxic or hazardous substances etc. End-point detection Is usually based on E/A.V maxima and on first or second derivatives In the case of microprocessor- and microcomputer-controlled processes, respectively. Table 13.2 lists a chronological selection of calculation methods applied to titration curves [46]. 

Principle. By means of potentiometric titration (in nonaqueous media) of a blend of sulfonic and sulfuric acids, it is possible to split the neutralization points corresponding to the first proton of sulfuric acid plus that of sulfonic acid, and to the second proton of sulfuric acid. The first derivate of the titration curve allows identification of the second points the corresponding difference in the volume of titrating agent is used as a starting point in the calculation method (Fig. 4). 

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