Titration, midpoint

Figure 21.5 Potential errors introduced by the assumption that Ar2+ is constant in the derivation of the linkage Eq. (21.30). The dependence of Ar2+ on [Mg2 1J was modeled by a polynomial fit to the solid gray data points in Fig. 21.4D. The polynomial was used in the integration of Eq. (21.30) to give expressions for In Kobs and 0 with [Mg2+]-dependent AT2 (in contrast to Eqs. (21.33) and (21.34), which assume AT2+ is constant). 9 is plotted for the calculated titration curve when the midpoint of the titration, [Mg2+]0, is 10 jiM (circles), 30 uA / (squares), or 100 li.M (diamonds). AT2+ (as used to calculate the displayed curves) at the titration midpoints (9 = 0.5) is 1.45, 2.30, and 2.73, respectively. The simulated data points have been fit to either a modified version of Eq. (21.34) that assumes the y-intercept of the curve has the value 6 = 0, 9 = 90 + (1 - 0o)([Mg2+]/[Mg2+]o)"/[l + ([Mg2+]/[Mg2+j0)"], or to an equation that allows a nonzero y-intercept, Eq. (21.35). The residuals of the fits are shown in the lower three panels closed symbols correspond to Eq. (21.34) and open symbols to Eq. (21.35). For the curve with a midpoint of [Mg2+]0 = 100 fiM, Eq. (21.35) could not be fit to the data because the value of Co became vanishingly small. The values of A IT. at the titration curve midpoints obtained from the modified Eq. (21.34) are 1.99, 2.39, and 2.73, in order ofincreasing [Mg2+]0. AT2+ obtained by fitting ofEq. (21.35) and application of Eq. (21.36) are 1.43 and 2.29 (10 and 30 fiM transition midpoints, respectively).

This reaction shows that Eh is related to the standard potential ( ° - 0.77 V) and the proportionality of Fe3+ to Fe2+ in the solution phase. A number of J5° values representing various reactions in soils are given in Tables 5.2 and 5.3. Note that Equation 5.16 is analogous to the Henderson-Hasselbalch equation. A plot of Eh versus (Fe3+/Fe3+ + Fe2+) would produce a sigmoidal line with midpoint ° (Fig. 5.2) (recall that the Henderson-Hasselbalch equation gives the pKa at the titration midpoint see Chapter 1). To the left of ° (midpoint in the jc axis E° = 0.77 V) the reduced species (e.g., Fe2+) predominates, whereas to the right of ° (midpoint in the x axis), the oxidized species (e.g., Fe3+) predominates. 

The two forms of the indicator, HIn and In, have different colors. The color of a solution containing an indicator, therefore, continuously changes as the concentration of HIn decreases and the concentration of In increases. If we assume that both HIn and In can be detected with equal ease, then the transition between the two colors reaches its midpoint when their concentrations are identical or when the pH is equal to the indicator s piQ. The equivalence point and the end point coincide, therefore, if an indicator is selected whose piQ is equal to the pH at the equivalence point, and the titration is continued until the indicator s color is exactly halfway between that for HIn and In. Unfortunately, the exact pH at the equivalence point is rarely known. In addition, detecting the point where the concentrations of HIn and In are equal maybe difficult if the change in color is subtle. 

As the titration begins, mostly HAc is present, plus some H and Ac in amounts that can be calculated (see the Example on page 45). Addition of a solution of NaOH allows hydroxide ions to neutralize any H present. Note that reaction (2) as written is strongly favored its apparent equilibrium constant is greater than lO As H is neutralized, more HAc dissociates to H and Ac. As further NaOH is added, the pH gradually increases as Ac accumulates at the expense of diminishing HAc and the neutralization of H. At the point where half of the HAc has been neutralized, that is, where 0.5 equivalent of OH has been added, the concentrations of HAc and Ac are equal and pH = pV, for HAc. Thus, we have an experimental method for determining the pV, values of weak electrolytes. These p V, values lie at the midpoint of their respective titration curves. After all of the acid has been neutralized (that is, when one equivalent of base has been added), the pH rises exponentially. 

The shapes of the titration curves of weak electrolytes are identical, as Figure 2.13 reveals. Note, however, that the midpoints of the different curves vary in a way that characterizes the particular electrolytes. The pV, for acetic acid is 4.76, the pV, for imidazole is 6.99, and that for ammonium is 9.25. These pV, values are directly related to the dissociation constants of these substances, or, viewed the other way, to the relative affinities of the conjugate bases for protons. NH3 has a high affinity for protons compared to Ac NH4 is a poor acid compared to HAc. 

FIGURE 2.14 The titration curve for phosphoric acid. The chemical formulas show the prevailing ionic species present at various pH values. Phosphoric acid (H3PO4) has three titratable hydrogens and therefore three midpoints are seen at pH 2.15 (p. i), pH 7.20 (p. 2). and pH 19.4 (p. 3). 

Redox titrations of the gray form monitored by optical spectroscopy indicate midpoint potentials of approx. +4 and -1-240 mV for centers I and II, respectively. 

Redox titrations monitored by visible and EPR spectroscopies show that the clusters have very different midpoint redox potentials approximately 0 mV for center I, and < - 300 mV for center II (139). 

This buffer region contains the midpoint of the titration, the point at which the amount of added OH" is equal to exactly half the weak acid originally present. In the current example, the solution at the midpoint contains 0.0375 mol each of acetic acid and acetate. Applying the buffer equation reveals the key feature of the midpoint ... 

At the midpoint of a titration of a weak acid by a strong base, the pH of the solution equals the p of the weak acid. 

In the long, flat buffer region, both B and its conjugate acid B are major species. The midpoint of the titration occurs in the buffer region. At this point, the pH of the solution is equal to the p. a of the conjugate acid of the base. 

The titration curve does not give directly. However, at the midpoint of the titration the concentration of Ep and EpH+ are identical. Use this information in the buffer equation to show that at the midpoint of the titration, the pH of the solution equals the p for EpH ... 

You may wonder why we did not use the titration curve to determine the pH at the stoichiometric point of the ephedrine titration, as we did to find the pH at the midpoint. Notice from Figure 18-6 that near the stoichiometric point, the pH changes very rapidly with added H3 O. At the stoichiometric point, the curve is nearly vertical. Thus, there Is much uncertainty in reading a graph to determine the pH at the stoichiometric point. In contrast, a titration curve is nearly fiat in the vicinity of the midpoint, minimizing uncertainty caused by errors in graph reading. 

Point A iies aiong the section of the titration curve known as the buffer region. Buffering action comes from the presence of a weak acid and its conjugate base as major species in solution. Moreover, Point A iies beyond the midpoint of the titration, which teiis us that more than half of the weak acid has been consumed. We represent this soiution with two moiecuies of H four ions of A, and four H2 O moiecuies ... 

Moving beyond the first stoichiometric point, the titration enters the second buffer region. Here, the major species are H A and its conjugate base,. The pH in this region is given by the buffer equation, using the p of H j4.". At the second midpoint, [H j4" ] = [j4 " ], and pH = p. Ta 2 For the titration of maleic acid, pH = 6.59 at the second midpoint. 

C18-0103. When a solution of leucine (an amino acid) in water is titrated with strong base, the pH before titration is 1.85, the pH at the midpoint of the titration is 2.36, and the pH at the stoichiometric point is 6.00. Determine the value of K. for leucine. 

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 5.15 A drawing of a titration curve for a weak acid pointing out the buffer region and midpoint. 

At the midpoint of the titration, the weak acid concentration equals the conjugate base concentration (half of the acid has been converted to the conjugate base) ... 

Table 5.1 gives commonly used examples of conjugate acid-base pair combinations and the pH range for which each is useful. This range corresponds to the pH range defined by the buffer region in the titration curve for each, and the middle of the range corresponds to the midpoint of each titration. 

To calculate gr from equation 11.15, it is necessary to know a, ft, V, gi, gf, 7j, Tf, and T. The values of a and ft are obtained as described. The volume of liquid inside the vessel can be calculated from the initial volume of titrate and the volume of titrant added. The temperature of the thermostatic jacket 7] must be independently measured. The values of g, and 7j, gf and Tf are usually taken at the midpoints of the fore and after periods, respectively, although e, refers to point b in the curve of figure 11.2. 

Figure 5.9 Example of a redox titration of nickel of hydrogenase from M. marburgensis. The amplitude of the Ni EPR signal is plotted against the measured redox potential. Half of the active sites in the enzyme solution is reduced at a redox potential (midpoint potential) of — 140 mV (at pH 6).The 2H /H2 redox couple has an of —354mV at this pH.The line through the points is a theoretical line assuming a midpoint potential of — 140 mV (Coremans et al. 1989).

Using the spectral data of Fig. 22, and similar data obtained for the nitrophorins in the absence of NO and in the presence of histamine, imidazole, or 4-iodopyrazole, Nernst plots such as that shown in the insert of Fig. 22 were constructed, and the midpoint potentials of the nitrophorins and their NO and histamine complexes were calculated. The results are summarized in Table IV, where they are compared to those obtained earlier for NPl (49, 50, 55). All potentials are expressed vs NHE (+205 mV with respect to the Ag/AgCl electrode used in the spectroelectrochemical titrations and the Nernst plot shown in the insert of Fig. 22). It can be seen that the reduction potentials of all four nitrophorins in the absence of NO or histamine are within 20-40 mV of each other. The reduction potentials of their NO complexes, however, differ significantly from each other. For example, the reduction potential of NP4-NO is about 350 mV more positive than that of NP4 in the absence of NO, as compared to a 430 mV shift for NPl upon binding NO, and the positive shifts for NP2—NO and NP3—NO are somewhat smaller (318 and 336 mV, respectively, at pH 7.5) 49, 50). These differences relate to the ratios of the dissociation constants for the two oxidation states, as discussed later. 

As NaOH is gradually introduced, the added OH-combines with the free H+ in the solution to form H20, to an extent that satisfies the equilibrium relationship in Equation 2-7. As free H+ is removed, HAc dissociates further to satisfy its own equilibrium constant (Eqn 2-8). The net result as the titration proceeds is that more and more HAc ionizes, forming Ac-, as the NaOH is added. At the midpoint of the titration, at which exactly 0.5 equivalent of NaOH has been added, one-half of the original acetic acid has undergone dissociation, so that the concentration of the proton donor, [HAc], now equals that of the proton acceptor, [Ac-]. At this midpoint a very important relationship holds the pH of the equimolar solution of acetic acid and acetate is ex-... 

FIGURE 2-17 The titration curve of acetic acid. After addition of each increment of NaOH to the acetic acid solution, the pH of the mixture is measured. This value is plotted against the amount of NaOH expressed as a fraction of the total NaOH required to convert all the acetic acid to its deprotonated form, acetate. The points so obtained yield the titration curve. Shown in the boxes are the predominant ionic forms at the points designated. At the midpoint of the titration, the concentrations of the proton donor and proton acceptor are equal, and the pH is numerically equal to the pAfa. The shaded zone is the useful region of buffering power, generally between 10% and 90% titration of the weak acid. 

This equation fits the titration curve of all weak acids and enables us to deduce a number of important quantitative relationships. For example, it shows why the pKa of a weak acid is equal to the pH of the solution at the midpoint of its titration. At that point, [HA] equals [A-], and... 

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