Titration curve 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... 

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... 

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... 

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

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. 

Figure S2 Enthalpy change as a function of concentration for titration of the aqueous solution of [C CgPyrrJBr (1.6 M) into water at 323.15 K. The data (squares) were fitted in a sigmoidal curve (filled curve), and the CMC was obtained as the zero of the second derivative (dotted curve, multiplied by 10).

Dilute 69 cm of purest syrupy phosphoric acid to 1 dm to obtain 1 M solution and by 10-fold dilution, prepare 0.1 M solution. Pipette 20.0 cm of the acid into a 250 cm conical flask. Calibrate the pH meter with a combined glass electrode and two buffer solutions at pH 4 and at 7. Dip the rinsed electrode into the flask, introduce a magnetic bar and add from a burette 0.1 M KOH solution in 2 cm aliquots, reading the pH on the meter when it settles. When the pH change becomes more rapid as the titration proceeds, decrease the volume down to 0.5 cm and then to 0.1 cm. After noting the first inflection point, continue with 2 cm additions of KOH and eventually to 0.1 cm addition until the second inflection point is reached. If a facility is available record the titration curve and preferably the derivative titration curve. 

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). 

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... 

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. 

Plot the titration curve in the vicinity of the end-point and its first and second derivatives from the following data and compare the end-point values ... 

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. 

In most cases, a curve is not drawn and the end point is taken as the milliliters (mL) used just when the color change takes place. There should be a half-drop of titrant difference between the change from one color to the next. In many cases, the color change is very light but distinctive. If the titration curve is plotted, then the end point can be determined by inspection or by taking the first or second derivative of the data to find the point of maximum slope. 

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. 

When the second derivative of (5.32) is calculated and set equal to zero, the inflection point of the titration curve is obtained [23, 24, 133, 134). It has been found that the theoretical titration error generally increases with decreasing sample concentration, with increasing value of the solubility product or of the dissociation constant, with increasing value of the dilution factor and with increasing concentration of the interferents. Larger errors are obtained with unsymmetrical titration reactions. The overall error is a combination of these factors the greatest effect is exerted by the sample concentration, a smaller one by the equilibrium constant and the interferents, and the smallest by dilution. To obtain errors below 1%, it must approximately hold that eg, > 10 2 i,K< 10 , < 10 to 10" and r < 0.3. 

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. 

We now turn our attention to details of precipitation titrations as an illustration of principles that underlie all titrations. We first study how concentrations of analyte and titrant vary during a titration and then derive equations that can be used to predict titration curves. One reason to calculate titration curves is to understand the chemistry that occurs during titrations. A second reason is to learn how experimental control can be exerted to influence the quality of an analytical titration. For example, certain titrations conducted at the wrong pH could give no discernible end point. In precipitation titrations, the concentrations of analyte and titrant and the size of Ksp influence the sharpness of the end point. For acid-base titrations (Chapter 11) and oxidation-reduction titrations (Chapter 16). the theoretical titration curve enables us to choose an appropriate indicator. 

Titration curves in Figure 7-6 illustrate the effect of reactant concentration. The equivalence point is the steepest point of the curve. It is the point of maximum slope (a negative slope in this case) and is therefore an inflection point (at which the second derivative is 0) ... 

The complete titration curve in Figure 11-1 exhibits a rapid change in pH near the equivalence point. The equivalence point is where the slope (dpH/dVf) is greatest (and the second derivative is 0, which makes it an inflection point). To repeat an important statement, the pH at the equivalence point is 7.00 only in a strong-acid-strong-base titration. If one or both of the reactants are weak, the equivalence point pH is not 7.00. 

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. 

Table 11 -3 Computation of first and second derivatives for a titration curve...

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. 

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. 

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. 

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. 

Firstly, ion exchange resins when hydrated generally dissociate to yield equivalent amounts of oppositely charged ions. Secondly, as with conventional aqueous acid or alkali solutions, resins in their acid or base forms may be neutralized to give the appropriate salt form. Finally, the degree of dissociation can be expressed in the form of an apparent equilibrium constant (or pK value) which defines the electrolyte strength of the exchanger and is usually derived from a theoretical treatment of pH titration curves. ... 

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