Calculating Redox Titration Curves

Anyone with a serious need to calculate redox titration curves should use a spreadsheet with a more general set of equations than we use in this section.5 The supplement at www.freeman.com/qca explains how to use spreadsheets to compute redox titration curves. 

Sketching a Redox Titration Curve As we have done for acid-base and complexo-metric titrations, we now show how to quickly sketch a redox titration curve using a minimum number of calculations. 

We will use standard electrode potentials throughout the rest of this text to calculate cell potentials and equilibrium constants for redox reactions as well as to calculate data for redox titration curves. You should be aware that such calculations sometimes lead to results that are significantly different from those you would obtain in the laboratory. There are two main sources of these differences (1) the necessity of using concentrations in place of activities in the Nernst equation and (2) failure to take into account other equilibria such as dissociation, association, complex formation, and solvolysis. Measurement of electrode potentials can allow us to investigate these equilibria and determine their equilibrium constants, however. 

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... 

Thus, = ( 0 /00 +) + 2 °(Sn +/Sn +). In practice, it is often easier to observe the equivalaice point than to calculate it If you must calculate the shape of a redox titration curve, the use of a spreadsheet program such as Excel is invaluable because of the multiple equilibrium equadons that must be solved. (See the text by Harris for excellent examples of spreadsheet calculations for redox titradons.)... 

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.

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. 

Calculate or sketch (or both) titration curves for the following (unbalanced) redox titration reactions at 25 °C. Assume that the analyte is initially present at a concentration of 0.0100 M and that a 25.0-mL sample is taken for analysis. The titrant, which is the underlined species in each reaction, is 0.0100 M. 

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. 

If both the half-reactions involved in a redox titration can be made to behave reversibly at a suitable electrode, the shapes of the titration curves should conform closely to the calculated values, though as pointed out above, the electrode potential reaches its equilibrium value more and more slowly with increasing dilution. 

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. 

T raditionally, titration curve calculations are described in terms of equations that are valid only for parts of the titration. Equations will be developed here that reliably describe the entire curve. This will be done first for acid-base titration curves. In following chapters, titration curves for other reaction systems (metal complexation, redox, precipitation) will be developed and characterized in a similar fashion. For all, graphical and algebraic means of locating the endpoints will be described, colorimetric indicators and how they function will be explained, and the application of these considerations to (1) calculation of titration errors, (2) buffo design and evaluation, (3) sharpness of titrations, and finally, (4) in Chapter 18, the use of titration curve data to the determination of equilibrium constants will be presented. 

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... 

This approach is effective not only for calculations involving individual solutions. The PBE and other balance equations apply to titration curve calculations as well. A single, easily derived, equation suffices to describe all points on a titration curve, be it for acid-base, complexometric, redox, or precipitation titrations. 

The concept of formal potentials has been developed for the mathematical treatment of redox titrations, because it was quickly realized that the standard potentials cannot be used to explain potentiometric titration curves. Generally, formal potentials are experimentally determined using equations similar to Eq. (1.2.24) because it is easy to control the overall concentrations of species in the two redox states. For calculating formal potentials it would be necessary to know the standard potential, all equilibrium constants of side reactions , and the concentrations of all solution constituents. In many cases this is still impossible as many equilibrium constants and the underlying chemical equilibria are still unknown. It is the great advantage of the concept of formal potentials to enable a quantitative description of the redox... 

Differential method (applicable to acid-base, precipitation and redox titrations). In. this method the differential curve is plotted directly instead of being calculated from the e.m.f.-volume graph. For the titration of solution AB with solution CD, X and Y are electrodes reversible to A, connected to the measuring instrument Y is enclosed in a tube which temporarily holds back a portion of AB (figure P.ll). 

O - back titration with ferricyanide. Closed symbols, 812 nm fluorescence - titration to lower redox potentials with dithionite, and - back titration with ferricyanide. The solid line is a calculated Nemst curve assuming a one-electron change and a midpoint potential of -168 mV. Redox potentials are vs. the normal hydrogen electrode. 

To confirm that a redox component regulates the apparent quantum yield of PSI we studied the dependence of PS I-activity on the redox-potential of the medium. Thy-lakoids were kept in the dark for 5-10 min and it was taken care, that the redox-potential of the medium was constant prior to the measurement. The plots of two experiments shown in figure 4 were fitted by a Nernst-curve of a titration with an one-electron step (n=l). The calculated midpoint-potential was +140m / (pH 7,8) in both experiments. 

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