Redox titration curve equivalence point

The data in the third column of Table 19-2 are plotted as curve B in Figure 19-3 to compare the two titrations. The two curves are identical for volumes greater than 25.10 mL because the concentrati ons of the two cerium species are identical in this region. It is also interesting that the curve for iron(Il) is symmetric around the equivalence point, but the curve for uranium(IV) is not. In general, redox titration curves are symmetric when the analyte and titrant react in a 1 1 molar ratio. 

Before we discuss redox titration curves based on reduction-oxidation potentials, we need to learn how to calculate equilibrium constants for redox reactions from the half-reaction potentials. The reaction equilibrium constant is used in calculating equilibrium concentrations at the equivalence point, in order to calculate the equivalence point potential. Recall from Chapter 12 that since a cell voltage is zero at reaction equilibrium, the difference between the two half-reaction potentials is zero (or the two potentials are equal), and the Nemst equations for the halfreactions can be equated. When the equations are combined, the log term is that of the equilibrium constant expression for the reaction (see Equation 12.20), and a numerical value can be calculated for the equilibrium constant. This is a consequence of the relationship between the free energy and the equilibrium constant of a reaction. Recall from Equation 6.10 that AG° = —RT In K. Since AG° = —nFE° for the reaction, then... 

A selected list of redox indicators will be found in Table 8.26. A redox indicator should be selected so that its if" is approximately equal to the electrode potential at the equivalent point, or so that the color change will occur at an appropriate part of the titration curve. If n is the number of electrons involved in the transition from the reduced to the oxidized form of the indicator, the range in which the color change occurs is approximately given by if" 0.06/n volt (V) for a two-color indicator whose forms are equally intensely colored. Since hydrogen ions are involved in the redox equilibria of many indicators, it must be recognized that the color change interval of such an indicator will vary with pH. 

Where Is the Equivalence Point In discussing acid-base titrations and com-plexometric titrations, we noted that the equivalence point is almost identical with the inflection point located in the sharply rising part of the titration curve. If you look back at Figures 9.8 and 9.28, you will see that for acid-base and com-plexometric titrations the inflection point is also in the middle of the titration curve s sharp rise (we call this a symmetrical equivalence point). This makes it relatively easy to find the equivalence point when you sketch these titration curves. When the stoichiometry of a redox titration is symmetrical (one mole analyte per mole of titrant), then the equivalence point also is symmetrical. If the stoichiometry is not symmetrical, then the equivalence point will lie closer to the top or bottom of the titration curve s sharp rise. In this case the equivalence point is said to be asymmetrical. Example 9.12 shows how to calculate the equivalence point potential in this situation. 

The scale of operations, accuracy, precision, sensitivity, time, and cost of methods involving redox titrations are similar to those described earlier in the chapter for acid-base and complexometric titrimetric methods. As with acid-base titrations, redox titrations can be extended to the analysis of mixtures if there is a significant difference in the ease with which the analytes can be oxidized or reduced. Figure 9.40 shows an example of the titration curve for a mixture of Fe + and Sn +, using Ce + as the titrant. The titration of a mixture of analytes whose standard-state potentials or formal potentials differ by at least 200 mV will result in a separate equivalence point for each analyte. 

It has been shown (Section 10.89) that the potential at the equivalence point is the mean of the two standard redox potentials. In Fig. 10.14, the curve shows the variation of the potential during the titration of 0.1 M iron(II) ion with... 

The titration is represented in Fig. 2.22 by plotting the Pt electrode potential versus the titration parameter k. BB is the voltage curve for titration of Fe2+ with Ce4+ and B B that for titration of Ce4+ with Fe2+ they correspond exactly to the pH curves BB and B B in Fig. 2.18, with the exception that the initial point in Fig. 2.22 would theoretically have an infinitely negative and an infinitely positive potential, respectively. In practice this is impossible, because even in the absence of any other type of redox potential there will be always a trace of Fe3+ in addition to Fe2+ and of Ce3+ in addition to Ce4+ present. Further, half way through the oxidation or reduction the voltage corresponds to the standard reduction potentials of the respective redox couples it also follows that the equivalence point is represented by the mean value of both standard potentials ... 

With a low constant current -1 (see Fig. 3.71) one obtains the same type of curve but its position is slightly higher and the potential falls just beyond the equivalence point (see Fig. 3.72, anodic curve -1). In order to minimize the aforementioned deviations from the equivalence point, I should be taken as low as possible. Now, it will be clear that the zero current line (abscissa) in Fig. 3.71 yields the well known non-faradaic potentiometric titration curve (B B in Fig. 2.22) with the correct equivalence point at 1.107 V this means that, when two electroactive redox systems are involved, there is no real need for constant-current potentiometry, whereas this technique becomes of major advantage... 

Again for the titration of Ce(IV) with Fe(II) we shall now consider constant-potential amperometry at one Pt indicator electrode and do so on the basis of the voltammetric curves in Fig. 3.71. One can make a choice from three potentials eu e2 and e3, where the curves are virtually horizontal. Fig. 3.74 shows the current changes concerned during titration at e1 there is no deflection at all as it concerns Fe(III) and Fe(II) only at e2 and e3 there is a deflection at A = 1 but only to an extent determined by the ratio of the it values of the Ce and Fe redox couples. The establishment of the deflection point is easiest at e2 as it simply agrees with the intersection with the zero-current abscissa as being the equivalence point in fact, no deflection is needed in order to determine this intersection point, but if there is a deflection, the amperometric method is not useful compared with the non-faradaic potentiometric titration unless the concentration of analyte is too low. 

In titrations involving 1 1 stoichiometry of reactants, the equivalence point is the steepest point of the titration curve. This is true of acid-base, complexometric, and redox titrations as well. For stoichiometries other than 1 1, such as 2Ag+ + CrO —> Ag2Cr04(s), the curve is not symmetric near the equivalence point. The equivalence point is not at the center of the steepest section of the curve, and it is not an inflection point. In practice, conditions are chosen such that titration curves are steep enough for the steepest point to be a good estimate of the equivalence point, regardless of the stoichiometry. 

The voltage change near the equivalence point increases as the difference between 6° of the two redox couples in the titration increases. The larger the difference in 6°, the greater the equilibrium constant for the titration reaction. For Figure 16-2, half-reactions 16-2 and 16-3 differ by 0.93 V, and there is a large break at the equivalence point in the titration curve. For Figure 16-3, the half-reactions differ by 0.47 V, so there is a smaller break at the equivalence point. 

The larger the difference in standard potential between titrant and analyte, the greater the break in the titration curve at the equivalence point. A redox titration is usually feasible if the difference between analyte and titrant is a 0.2 V. However, the end point of such a titration is not very sharp and is best detected potentiometrically. If the difference in formal potentials is a 0.4 V. then a redox indicator usually gives a satisfactory end point. 

Another specialized form of potentiometric endpoint detection is the use of dual-polarized electrodes, which consists of two metal pieces of electrode material, usually platinum, through which is imposed a small constant current, usually 2-10 /xA. The scheme of the electric circuit for this kind of titration is presented in Figure 4.1b. The differential potential created by the imposition of the ament is a function of the redox couples present in the titration solution. Examples of the resultant titration curve for three different systems are illustrated in Figure 4.3. In the case of two reversible couples, such as the titration of iron(II) with cerium(IV), curve a results in which there is little potential difference after initiation of the titration up to the equivalence point. Hie titration of arsenic(III) with iodine is representative of an irreversible couple that is titrated with a reversible system. Hence, prior to the equivalence point a large potential difference exists because the passage of current requires decomposition of the solvent for the cathode reaction (Figure 4.3b). Past the equivalence point the potential difference drops to zero because of the presence of both iodine and iodide ion. In contrast, when a reversible couple is titrated with an irreversible couple, the initial potential difference is equal to zero and the large potential difference appears after the equivalence point is reached. 

When the equivalence point is reached, the Fe2+ will have been totally consumed (the large equilibrium constant ensures that this will be so), and the potential will then be controlled by the concentration ratio of Ce3+/Ce4+. The idea is that both species of a redox couple must be present in reasonable concentrations for a concentration to control the potential of an electrode of this kind. If one works out the actual cell potentials for various concentrations of all these species, the resulting titration curve looks much like the familiar acid-base titration curve. The end point is found not by measuring a particular cell voltage, but by finding what volume of titrant gives the steepest part of the curve. 

Any titration involves the progressive change of the activities (or concentrations) of the titrated and titrating species and, in principle, can be done potentiometrically. However, for an accurate determination it is necessary that there is a fairly rapid variation in equilibrium potential in the region of the equivalence point. Useful applications are redox, complexation, precipitation, acid-base titrations, etc. From the titration curve it is possible to calculate values of the formal potentials of the titrated and titrating species, as explained below. 

A titration curve may for convenience be considered to consist of three portions the region before the equivalence point, the equivalence point, and the region beyond the equivalence point. At all points except at the beginning before any titrant has been added, two redox couples are present, corresponding to the sample and the titrant. In the region before the equivalence point, the potential is calculated conveniently from the known concentration ratio of the sample redox couple. After the equivalence point the concentration ratio of the titrant redox couple is known from the stoichiometry. At the equivalence point both the sample and titrant redox couples are present in the stoichiometric ratio. 

Construct a coulometric titration curve of 100.0 mL of a 1 M H2SO4 solution containing Fe(ll) titrated with Ce(lV) generated from 0.075 M Ce(lll). The titration is monitored by potentiometry. The initial amount of Fe(II) present is 0.05182 mmol. A constant current of 20.0 mA is used. Find the time corresponding to the equivalence point. Then, for about 10 values of time before the equivalence point, use the stoichiometry of the reaction to calculate the amount of Fe produced and the amount of Fe + remaining. Use the Nemst equation to find the system potential. Find the equivalence point potential in the usual manner for a redox titration. For about 10 times after the equivalence point, calculate the amount of Ce " produced from the electrolysis and the amount of Ce + remaining. Plot the curve of system potential versus electrolysis time. 

We can use our understanding of redox equihbria to describe titration curves for redox titrations. The shape of a titration curve can be predicted from the E° values of the analyte half-reaction and the titrant half-reaction. Roughly, the potential change in going from one side of the equivalence point to the other will be equal to the difference in the two values the potential will be near E° for the analyte half-reaction before the equivalence point and near that of the titrant half-reaction beyond the equivalence point. 

So, redox indicators will have a transition range over a certain potential, and this transition range must fall within the steep equivalence point break of the titration curve. The redox indicator reaction must be rapid, and to use terms of the electrochemist, it must be reversible.Jf the reaction is slow or is irreversible (slow rate of electron transfer), the color change will be gradual and a sharp end point will not be detected. 

The shape of the amperometric titration curve in this case, where both the titrant and the substance titrated undergo reversible redox reactions, is illustrated in Figure 3.21A. In the case where the substance titrated does not have a reversible voltammetric wave, the titration curve will have the shape illustrated in Figure 3.2IB. Prior to the equivalence point, the applied voltage is too small to cause both oxidation and reduction of the redox couple of the substance titrated. If the titrant has an irreversible wave, the titration curve will look like that in Figure 3.21C. This type of titration is commonly called a dead-stop titration, because the indicator current falls to zero at the equivalence point. 

Selected points along the titration curve are calculated in a manner that closely resembles those we used in the last two chapters. With oxidation-reduction titrations, the method is even simpler when we recognize that on either side of the equivalence point there is an excess of one of the two redox couples, allowing us to calculate the concentration ratio of oxidized to reduced forms of the substance being titrated or of the titrant. Knowing... 

Figure 4 Amperometric titration curves for redox reactions on dual polarizable electrodes (A) reversible half-reactions for titrant and sample (B) reversible half-reaction for sample only and (C) reversible half-reaction for titrant only, e.p., equivalent point (Reprinted with permission from Peters GD, Hayes JM, and Hieftje GM (1974) Chemical Separations and Measurements Brooks/Cole, a division of Thomson Learning www.thomson-rights.com. fax 800 730-2215.)...

The larger the difference in standard potential between titrant and analyte, the sharper the break in the titration curve at the equivalence point. A redox titration is... 

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