Calculations, Titration curves

By now you are familiar with our approach to calculating titration curves. The first task is to calculate the volume of Ag+ needed to reach the equivalence point. The stoichiometry of the reaction requires that... 

At first we determined, by means of the DVP method, ifTMAX of 2,4-dinitro-phenolate, 2,5-dinitrophenol picrate, acetate and benzoate, which lay between 10 3 and 10 5. Next, separate potentiometric titrations of 2,5-dinitrophenol and picric acid were carried out on the basis of the previously known (see above) ptfax = 6.5 and P hx2- = 100 for 2,5-dinitrophenol and p.fiTHX = 3.0 for picric acid, we calculated titration curves for estimated values of 0 and obtained, for the best fit between the experimental and calculated curves, K o = 10 21 for both 2,5-dinitrophenol and picric acid. In both instances changing fTMA0H for 1 to 10 6 did not alter the calculated titration curve. Finally, for potentiometric titrations of other acids with TMAOH while using / TMAX values from DVP results, in addition to Kn 0 = 10 21, we obtained the best fit between the experimental and calculated curves again when pifbenzoic acid = 1 (see Fig. 4.12)... 

In textbooks of computational chemistry you will invariably find examples calculating the pH = - lg [H+]/(mol/l)> in weak acid - strong base or strong acid - weak base solutions. Indeed, these examples are important in the study of acids, bases and of complex formation, as well as for calculating titration curves. Following (ref. 24) we consider here the aquous solution that contains a weak tribasic acid H A and its sodium salts NaH, Na HA and Na A in known initial concentrations. The dissociation reactions and equilibrium relations are given as follows. 

Fig. 3.1 Calculated titration curves of a strong acid and weak acids of various pKa values with a strong base. In the solvent of pffsH = 24 and at the acid concentration of 10 2 M. The effect of activity coefificent and that of dilution were neglected. [The dashed curve is for the case of p/CSH = 14 (water).]...

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. 

FigurB 11-1 Calculated titration curve, showing how pH changes as 0.100 0 M HBr is added to 50.00 mL of 0.020 00 M KOH. The equivalence point is also an inflection point.

Figure 77-2 Calculated titration curve for the reaction of 50.00 mL of 0.020 00 M MES with 0.100 0 M NaOH. Landmarks occur at half of the equivalence volume (pH = pKa) and at the equivalence point, which is the steepest part of the curve.

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).

Fig- 9 Solubilisation of (a) Arginine, and (b) phenylalanine in Reversed Micelles. The Solid Lines are Calculated Titration Curves Based on the pKa s for the Amino Acids. 

Fig. 15.4. Titration data from Tuominen (1967) for Cladonia alpestris, depicted as a function of pH versus concentration of added titrant. The closed circles represent forward titration data, while open circles stand for reversed titration data points. The upper curve is a calculated titration curve in pure water. The shaded area denotes the extent of pH buffering capacity exhibited by the lichen, relative to a non-buffering solution of pure water.

Figure 11-5 shows plots of calculated titration curves of 10 M nickel(II), iron(III), and calcium(II) at various pH values in buffers of ammonia-ammonium ion at a total buffer concentration of 0.1 M. Before the end point, the nickel curves at... 

In the Nemst equation only the ratio of [red]/[ox] appears, and so an important property of the calculated titration curve for Reaction (15-4) is that the theoretical shape is independent of the concentration of reactants. Thus the sharpness of the titration is theoretically unaffected by dilution. Note, however, that (15-4) is not perfectly general, because the simple relation of to n for reactants and products is not always valid. Consider, for example, the reaction... 

In calculating titration curves, separate equations for different regions of the curve ("before the equivalence point", "at the equivalence point", "after the equivalence point", etc.) are often employed. This section illustrates how to use a single "master" equation to calculate points on a titration curve. Instead of calculating pH as a function of the independent variable V, it is convenient to use pH as the independent variable and V as the dependent variable. The species distribution at a particular pH value is calculated from the [H ], and the volume of titrant required to produce that amount of each species is calculated. For example, in the titration of a weak monoprotic acid HA, we can calculate the concentration of A at a particular pH and then calculate the number of moles of base required to produce that amount of A . In general (J -j) moles of base per mole of acid are required to produce the species HjA from the original acid species HjA. 

There are a number of computer programs available for the determination of stability constants from pH titration data. The most general of these perform a least-squares fit of the data to a calculated titration curve. The programs are able to handle protonated complexes, polynuclear systems, etc. In this example least-squares curve fitting is applied to a somewhat simpler case, a polyprotic acid in which the equilibria overlap extensively. The method is that used in the... 

In following hand calculations of titration results we will ignore activity coefficients to avoid excessively complicating the calculations. Titration curves are typically plotted in two ways. The plot is usually either (1) of the volume of a titrant acid or base added (Vg) versus the pH of the solution or (2) of the concentration of acid or base added (Q or C ) versus the solution pH. The plots have a similar appearance. We will derive algebraic expressions for each type of plot. The advantage of or Cfi versus pH plots and mathematical expressions is that they can be used to derive the buffer... 

We will hot constract a diprotic titration curve here, but if you want a good mental exercise, try it You just can t make the simplifying assumptions that we can usually use with monoprotic acids that are sufficiently weak or not too dilute. See your CD, Chapter 8, for auxiliary data for the spreadsheet calculation of the titration curve for 50.00 mL 0.1000 M H2C1O4 versus 0.1000 M NaOH. You can download that and enter the Kai and Kai values for other diprotic acids and see what their titration curves look like. Try, for example, maleic acid. For the calculations, we used the more exact equations mentioned above for the initial pH, the first buffet zone, and the first equivalence point. We did not use the quadratic equation for the second equivalence point since Cr04 is a quite weak base (Kbi = 3.12 X 10 ). See Ref. 8 for other examples of calculated titration curves. 

Ionic equilibria in solvents of low relative permittivity have been described in the literature and equations for calculating titration curves have also been proposed. The equations involve the partial dissociation of the acids, bases, and salts in these media, and the effect of the activity coefficients. The equations can be applied to pH and titration curve computation in solvents such as t-butyl and isopropyl alcohols, ethylene diamine, pyridine, or tetrahydrofuran. 

In Sections 10-1 to 10-3, we calculated titration curves because they helped us understand the chemistry behind a titration curve. Now we will see how a spreadsheet decreases the agony and mistakes of titration calculations. First we must derive equations relating pH to volume of titrant for use in the spreadsheet. 

Figure 11-2 Calculated titration curves for three different diprotic acids, H2A. For each curve, 50.0 ml of 0.020 0 M H2A are titrated with 0.100 M NaOH. Lowest curve jpK = 4.00 and p/C 2 = 6.00. Middle curve = 4.00 and p/C 2 = 8.00. Upper curve = 4.00 and...

Figure 11-3 Calculated titration curve for 50.0 mL of 0.020 0 M Na2C03 titrated with 0.100 M HCl.

Figure 7-2. Calculated titration curve and its Gran plot for 0.100 M NH4 with NaOH, 0.100 M. Here F is the acid, or the base Gran function.

A calculated titration curve of oxalic acid titrated with a solution of sodium hydroxide... 

A calculated titration curve for oxalic acid is shown at the right. Oxalic acid has pK values of 1.27 and 4.27. Therefore the buffer regions will be centered at about pH 1.3 and... 

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