Equivalence point derivatives

Derivative methods work well only when sufficient data are recorded during the sharp rise in plT occurring near the equivalence point. This is usually not a problem when the titration is conducted with an automatic titrator, particularly when operated under computer control. Manual titrations, however, often contain only a few data points in the equivalence point region, due to the limited range of volumes over which the transition in plT occurs. Manual titrations are, however, information-rich during the more gently rising portions of the titration curve before and after the equivalence point. 

Derive a general equation for the electrochemical potential at the equivalence point for the titration of Fe + with Mn04 the reaction 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). 

The relevant results are collected in Table 15.5, as are also the calculated values for the first derivative AE/AV (millivolt mL-1) and the second derivative A2 /AF2 it is clear that for locating the end point, only the experimental figures in the vicinity of the equivalence point are required. It is convenient, and simplifies the calculations, if small equal volumes of titrant are added in the neighbourhood of the end point, but this is not essential. 

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). 

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. 

We may compare this, for example, with the equation at an equivalent point in Ingle and Crouch s development [3] (taking that as a typical derivation) ... 

After the equivalence point, the pH is determined directly from the concentration of excess NaOH since CH3COOH is now the limiting reactant. In the presence of the strong base, the effect of the weak base, CH3COO, derived from the salt is negligible. 

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. 

Derived structures may also be formed by the ordered introduction of vacant sites. As an example, consider the hP3-CdI2 type structure (see Chapter 7) which can be related to the hP4-NiAs type structure in which the set of equivalent points 0,0,0 and 0, 0, M is considered as being subdivided into two groups (each of one site) 0,0,0 (occupied by one atomic species) and 0, 0, M (vacant). We can therefore regard the hP3-CdI2 type structure as a defect derivative form of the hP4-NiAs type (see 7.4.2.4.3). Similar considerations maybe extended to include (besides substitution and subtraction) ordered addition of atoms. In this case stuffed or filled-up derivative... 

Derivative plots such as that shown in Figure 4.4 can greatly increase the accuracy of the end point determination, provided that a sufficient amount of data is obtained. We need to note that the rate of change of emf with volume V is often very large near the equivalence point, and so it is easy to miss some of the data. 

As it will be discussed, while three maxima of the first derivative are observed, the second one is a consequence of the applied numerical method. Using the second derivative values in the last column, local inverse linear interpolation gives V = 3.74 ml and V = 7.13 ml for the two equivalence points. We will see later on how the false end point can be eliminated. 

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. 

Alternatively, we can obtain the equivalent partial derivative identities of Table 5.2 for the comer variables. For example, S sits at the tail (negative) end of the arrow pointing to T, considered as a variable of either A (at constant V) or of G (at constant P), the two edges that meet at this comer. We can therefore write S as either of the partial derivatives... 

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. 

Table 11-6 gives useful equations derived by writing a charge balance and substituting fractional compositions for various concentrations. For titration of the diprotic acid, H2A, ct> is the fraction of the way to the first equivalence point. When = 2, we are at the second equivalence point. It should not surprise you that, when cj> = 0.5, pH = pAj and, when = 1.5, pH pAT2. When = 1, we have the intermediate HA " and pH j(pAj -I- pA j). 

Behavior of derivatives at the equivalent point First derivative ApH/A V has greatest magnitude Second derivative A( ApH/AV)/AV = 0... 

At the Equivalence Point After addition of 40.0 mL of 0.100 M NaOH, we have added (40.0 mL)(0.100 mmol/mL) = 4.00 mmol of NaOH, which is just enough OH- to neutralize all the 4.00 mmol of HC1 initially present. This is the equivalence point of the titration, and the pH is 7.00 because the solution contains only water and NaCl, a salt derived from a strong base and a strong acid. 

The equivalence point was found by CV to correspond to the interaction of one ferrocenyl branch per equiv. anion in this series of dendrimers. The 9- and 18-amidoferrocenyl dendrimers (resp. 9-Fc and 18-Fc) were compared to monomeric (1-Fc) and trinuclear (3-Fc) amidoferrocenyl derivatives (Fig. 2), and, as indicated by the above data, a strongly positive dendritic... 

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. 

The second derivative (Fig. 13.26). The equivalence point is where the curve crosses the V -axis. 

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.

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