First derivative titration

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... 

Titration curves for a weak acid with 0.100 M NaOH—(a) normal titration curve (b) first derivative titration curve ... 

A first derivative titration can be used to prepare a Gran plot. 

Potentiometry, Fig. 3 Potentiometric titration curves, (a) typical plot of E versus volume, (b) typical first derivative titration curve, (c) typical second derivative titration curve... 

Normal Titration First Derivative Second Derivative ... 

The second derivative of a titration curve may be more useful than the first derivative, since the end point is indicated by its intersection with the volume axis. The second derivative is approximated as A(ApH/AV)/AV, or A pH/AV. For the titration data in Table 9.5, the initial point in the second derivative titration curve is... 

The following data were collected with an automatic titrator during the titration of a monoprotic weak acid with a strong base. Prepare normal, first-derivative, second-derivative, and Gran plot titration curves for this data, and locate the equivalence point for each. 

Schwartz has published some hypothetical data for the titration of a 1.02 X ICr" M solution of a monoprotic weak acid (pXa = 8.16) with 1.004 X ICr M NaOH. " A 50-mL pipet is used to transfer a portion of the weak acid solution to the titration vessel. Calibration of the pipet, however, shows that it delivers a volume of only 49.94 ml. Prepare normal, first-derivative, second-derivative, and Gran plot titration curves for these data, and determine the equivalence point for each. How do these equivalence points compare with the expected equivalence point Comment on the utility of each titration curve for the analysis of very dilute solutions of very weak acids. 

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). 

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. 

In routine analysis, often a one-dimensional so-called end-point titration can be automatically carried out up to a pre-set pH or potential value and with a previously chosen overall titration velocity in order to avoid overshoot, the inflection point should be sufficiently sharp and the titrant delivery must automatically diminish on the approach to that point in order to maintain equilibrium, and stop in time at the pre-set value. For instance, the Metrohm 526 end-point titrator changes both the dosing pulse length and its velocity by means of a pulse regulator in accordance with the course of the titration curve in fact, the instrument follows the titration two-dimensionally, but finally reports only a one-dimensional result. The Radiometer ETS 822 end-point titration system offers similar possibilities. However, automated titrations mostly represent examples of a two-dimensional so-called eqilibrium titration, where the titration velocity is inversely proportional to the steepness of the potentiometric titration curve hence the first derivative of the curve can usually also be recorded as a more accurate means of determining the inflection... 

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

The volume of titrant added at the equivalence point of a titration can be accurately determined by plotting the first and second derivatives of the titration curve. A first derivative is a plot of the rate of change of the pH, ApH, vs. milliliters of titrant, and the second derivative is a plot of the rate of change of the first derivative, A(ApH), vs. milliliters of titrant. The plot in the center is the first derivative of the titration curve on the left, and the plot on the right is the second derivative. The rate of change of the curve on the left is a maximum at the midpoint of the inflection point, so the maximum on the first derivative coincides with this point, which is the equivalence point of the titration. Similarly, the rate of change is zero at the maximum of the curve in the center, so the equivalence point is also the point where the second derivative crosses zero. Thus, the equivalence point is the milliliters of titrant at the peak of the first derivative and the milliliters of titrant at the point where the line crosses zero for the second derivative. The second derivative provides the most precise measurement of the equivalence point. 

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. 

A first-derivative plot of a potentiometrie titration curve. 

In potentiometric titration a voltage is obtained from an electrode that is sensitive to an ionic species such as H-jO+, i.e., the pH of the solution in this case. We will consider the titration of the mixture of a strong acid (HC1) and a weak acid (CJ+jCOOH) with NaOH (ref. 10). As 2 ml volumes of the base are given to the acidic solution, the pH increases and when one of the acids is neutralized the pH changes very rapidly by a small addition of NaOH. We want to find these maximum points of the first derivative of the titration curve. In the following main program the DATA lines contain 32 data pairs, each consisting of the volume of the added NaOH in ml and the measured pH. 

Our goal is again to find the maxima of the smoothed first derivative of the titration curve first studied in Example 4.1.3. Recall that the discrete... 

Figure 11-6 (d) Experimental points in the titration of 1.430 mg of xylenol orange, a hexaprotic acid, dissolved in 1.000 mL of aqueous 0.10 M NaN03. The titrant was 0.065 92 M NaOH. (fc>) The first derivative, ApH/A V, of the titration curve, (c) The second derivative, A(ApH/A V)/AV, which is the derivative of the curve in panel b. Derivatives for the first end point are calculated in Table 11-3. End points are taken as maxima in the derivative curve and zero crossings of the second derivative. 

With the apparatus in Figure 16-1. we measure electrode potential. , versus volume of titrant, V, during a redox titration. The end point is the maximum of the first derivative, AE/AV, or the zero crossing of the second derivative. A(AE/AV)/AV (Figure 11-6). 

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. 

Potentiometric titration curves normally are represented by a plot of the indicator-electrode potential as a function of volume of titrant, as indicated in Fig. 4.2. However, there are some advantages if the data are plotted as the first derivative of the indicator potential with respect to volume of titrant (or even as the second derivative). Such titration curves also are indicated in Figure 4.2, and illustrate that a more definite endpoint indication is provided by both differential curves than by the integrated form of the titration curve. Furthermore, titration by repetitive constant-volume increments allows the endpoint to be determined without a plot of the titration curve the endpoint coincides with the condition when the differential potentiometric response per volume increment is a maximum. Likewise, the endpoint can be determined by using the second derivative the latter has distinct advantages in that there is some indication of the approach of the endpoint as the second derivative approaches a positive maximum just prior to the equivalence point before passing through zero. Such a second-derivative response is particularly attractive for automated titration systems that stop at the equivalence point. 

Figure 4.2 Potentiometric titration curves (a) experimental titration data (b) first derivative of curve a (c) second derivative of curve a.

In addition to these two versions of autotitrators, titrators have been developed that take advantage of the change in potential per increment of titrant added. Such instruments terminate the addition of titrant when the first derivative of potential relative to volume reaches a maximum, or when the second... 

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.

Figure 25. Direct titration of acidic groups on a carbon black. The curves represent the highresolution first derivatives of the titration curves. The ordinate scales are identical for both experiments (offset).

Differential potentiometric titration — means the experimental recording of the first derivative of potential over volume of added titrant of a -> potentiometric titration curve. This can be achieved with a -> retarded electrode as developed by - Maclnnes. In a broader sense this term also covers the mathematical derivation of potentiometric titration curves. 

When a - potentiometric titration ought to be followed by recording the first derivative this can be achieved by... 

Figure 9-10. First derivative of titration data, near the end-point.

The spreadsheet shown in Figure 9-10 uses pH titration data to illustrate the calculation of the first derivative of a data set. 

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