Titration first-derivative method

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. 

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. 

Figure 12.20 A typical redox titration curve for hydrogen peroxide measurement. The left ordinate is the millivolt reading from the redox electrode the right ordinate is the first derivative of the millivolt curve. Volume (mL) of titrant (permanganate) is shown on the abscissa. The endpoint shown is calculated with the first derivative method—the maximum of the pink curve. Used with permission from the author.

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... 

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). 

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. 

It must be realized that the constant current (-1) chosen virtually determines a constant titration velocity during the entire operation hence a high current shortens the titration time, which is acceptable at the start, but may endanger the establishment of equilibrium of the electrode potentials near the titration end-point in an automated potentiometric titration the latter is usually avoided by making the titration velocity inversely proportional to the first derivative, dE/dt. Now, as automation of coulometric titrations is an obvious step, preferably with computerization (see Part C), such a procedure can be achieved either by such an inversely proportional adjustment of the current value or by a corresponding proportional adjustment of an interruption frequency of the constant current once chosen. In this mode the method can be characterized as a potentiometric controlled-current coulometric titration. 

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.

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. 

FIG. 3.7 Inflection point method (first 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. 

In a typical potentiometric titration, the initial addition of a small amount of titrant produces little change in the e.m.f. of the cell (figure P.5), since this depends on the fraction of a particular ion removed. Towards the equivalence point, however, the fraction of ion removed by a constant amount increases rapidly, and this is reflected by a rapid change in e.m.f. Above the equivalence point the curve again flattens out. Graphical procedures for the exact location of the equivalence point are (a) method with first derivative (figure P.6a), (b) method with... 

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... 

In an early work by Mertz and Pettitt, an open system was devised, in which an extended variable, representing the extent of protonation, was used to couple the system to a chemical potential reservoir [67], This method was demonstrated in the simulation of the acid-base reaction of acetic acid with water [67], Recently, PHMD methods based on continuous protonation states have been developed, in which a set of continuous titration coordinates, A, bound between 0 and 1, is propagated simultaneously with the conformational degrees of freedom in explicit or continuum solvent MD simulations. In the acidostat method developed by Borjesson and Hiinenberger for explicit solvent simulations [13], A. is relaxed towards the equilibrium value via a first-order coupling scheme in analogy to Berendsen s thermostat [10]. However, the theoretical basis for the equilibrium condition used in the derivation seems unclear [3], A test using the pKa calculation for several small amines did not yield HH titration behavior [13],... 

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 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. 

Only one kinetic study exists on initiation of methacrylate polymerization by a sodium compound. The initiator was the disodium oligomer ( tetramer ) of a-methylstyrene and polymerization was investigated at 25°C in toluene in presence of 0.05—0.2 mole fraction of tetrahydrofuran [181]. An internal first order disappearance of monomer was observed, the first order coefficient being directly proportional to active chain and tetrahydrofuran concentrations. The rate coefficients evaluated, e.g. fep = 3.1—13 X 10 1 mole sec at various tetrahydrofuran concentrations, are much lower than those for lithium initiators. They were, however, evaluated using a methyl iodide titration technique to estimate the active chain concentration. In view of the reactivity of tritiated acetic acid with many short chains which are clearly not active in chain propagation, there must be suspicion of similar behaviour with methyl iodide. If this happens, the active chain concentration would be over-estimated and the derived fep value would be too low. Unfortunately no molecular weights of the precipitable polymer were determined, so that it is impossible to check on active chain concentration using this alternative method. 

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]

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