Calculating Titration Curves with Spreadsheets

Titrating a Weak Acid with a Strong Base 

Consider the titration of a volume Va of acid HA (initial concentration Ca) with a volume Vb of NaOH of concentration Cb. The charge balance for this solution is 

Now we use the fractional composition equations from Chapter 10. Equation 10-18 told us that 

Fraction of titration for CbVb weak acid by strong base CaVa 

At last Equation 11-9 is really useful. It relates the volume of titrant (Vh) to the pH and a bunch of constants. The quantity () , which is the quotient CbVb/CaVa, is the fraction of the way to the equivalence point, V e. When j = I, the volume of base added, Vb, is equal to Ee. Equation 11-9 works backward from the way you are accustomed to thinking, because you need to put in pH (on the right) to get out volume (on the left). Let me say that again We put in a concentration of H1 and get out the volume of titrant that produces that concentration. 

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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. 

The approach that we have worked out for the titration of a monoprotic weak acid with a strong base can be extended to reactions involving multiprotic acids or bases and mixtures of acids or bases. As the complexity of the titration increases, however, the necessary calculations become more time-consuming. Not surprisingly, a variety of algebraic and computer spreadsheet approaches have been described to aid in constructing titration curves. 

Eventually, we will derive a single, unified equation for a spreadsheet that treats all regions of the titration curve. To understand the chemistry of the titration, it is sensible to break the curve into three regions described by approximate equations that are easy to use with a calculator. 

By now you should understand the chemistry that occurs at different stages of a precipitation titration, and you should know how to calculate the shape of a titration curve. We now introduce spreadsheet calculations that are more powerful than hand calculations and less prone to error. If a spreadsheet is not available, you can skip this section with no loss in continuity. Consider the addition of liters of cation M+ (whose initial concentration is C ) to liters of solution containing anion X- with a concentration C%. 

Equation 7-18 relates the volume of added M+ to [M+], [X-], and the constants V7, C, and CjJ,. To use Equation 7-18 in a spreadsheet, enter values of pM and compute corresponding values ofVM, as shown in Figure 7-10 for the iodide titration of Figure 7-7. This is backward from the way you normally calculate a titration curve in which VM would be input and pM would be output. Column C of Figure 7-10 is calculated with the formula [M+l = 10 pm, and column D is given by [X-] = k sp/[M+]. Column E is calculated from Equation 7-18. The first input value of pM (15.08) was selected by trial and error to produce a small V You can start wherever you like. If your initial value of pM is before the true starting point, then VM in column E will be negative. In practice, you will want more points than we have shown so that you can plot an accurate titration curve. 

Figure 11-11 Spreadsheet that uses Equation 11-9 to calculate the titration curve for 50 mL of the weak acid 0.02 M MES (pKa = 6.27) treated with 0.1 M NaOH. We provide pH as input in column B and the spreadsheet tells us what volume of base is required to generate that pH. 

For a titration with EDTA, you can follow the derivation through and find that the formation constant, Kf, should be replaced in Equation 12-11 by the conditional formation constant, K, which applies at the fixed pH of the titration. Figure 12-12 shows a spreadsheet in which Equation 12-11 is used to calculate the Ca2+ titration curve in Figure 12-11. As in acid-base titrations, your input in column B is pM and the output in column E is volume of titrant. To find the initial point, vary pM until V, is close to 0. 

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. 

Figure 20-11 illustrates a portion of a spreadsheet for the calculation of the titration curve of 2.500 mmol of a weak acid (pXa = 5) with 0.1000 M strong base. The volume required to obtain a given pH value was calculated for pH values from 3 to 12 in increments of 0.20. The formula used to calculate V in cell C9 is... 

The titration curve of iron(II) with cerium(IV) appears as A in Figure 19-3. This plot resembles closely the curves encountered in neutralization, precipitation, and complex-formation titrations, with the equivalence point being signaled by a rapid change in the ordinate function. A titration involving 0.00500 M iron(II) and 0.01000 M cerium(IV) yields a curve that for all practical purposes is identical to the one we have derived, since the electrode potential of the system is independent of dilution. A spreadsheet to calculate iisysiem a function of the volume of Ce(lV) added is shown in Figure 19-4. 

Figure 1 9-4 Spreadsheet and plot for titration of 50.00 mL of 0.0500 M Fe " with 0.1000 M Ce. Prior to the equivalence point, the system potential is calculated from the and Fe + concentrations. After the equivalence point, the Ce and Ce + concentrations are used in the Nernst equation. The Fe concentration in cell B7 is calculated from the number of millimoles of Ce added, divided by the total volume of solution. The formula used for the first volume is shown in documentation cell A21. In cell Cl, [Fe- ] is calculated as the initial number of millimoles of Fe present, minus the number of millimoles of Fe formed, divided by the total solution volume. Documentation cell A22 gives the formula for the 5.00-mL volume. The system potential prior to the equivalence point is calculated in cells F7 F12 by using the Nernst equation, expressed for the first volume by the formula shown in documentation cell A23. In cell F13, the equivalence-point potential is found from the average of the two formal potentials, as shown in documentation cell A24. After the equivalence point, the Ce(lll) concentration (cell D14) is found from the number of millimoles of Fe- initially present divided by the total solution volume, as shown for the 25.10-mL volume by the formula in documentation cell D21. The Ce(IV) concentration (El 4) is found from the total number of millimoles of Ce(lV) added, minus the number of millimoles of Fe + initially present, divided by the total solution volume, as shown in documentation cell D22. The system potential in cell FI4 is found from the Nernst equation as shown in documentation cell D23. The chart is then the resulting titration curve.

Prepare a spreadsheet to construct the titration curve for the titration of 50 mL 0.05 M HOAc with 0.05 M NaOH. First enter the absolute values for the volume (cell B2) and concentration (cell D2) of HOAc, the concentration of NaOH (cell F2), Ky, (cell Gl), and (cell H2). These will be used in the following formulas to calculate the pH as a function of volume of NaOH added. ... 

See Problem 21 for a spreadsheet calculation of the Ca-EDTA titration curve in Figure 9.3 at pH 10. As with calculated acid-base titration curves, the calculations here break down very near the equivalence point due to simplifying assumptions we have made. 

Prepare a spreadsheet to plot the titration curve of 100 mL 0.1 M chloride titrated with 0.1 M silver nitrate (Figure 11.1). See your CD for a suggested setup. Use the spreadsheet to change the concentrations of chloride and silver (e.g., 0.2 M each, 0.05 M each), and notice how the titration curve changes. Note that there is a limit to how low the concentrations can go in these calculated plots because eventually the solubility of the AgCl at 99.9 and 100.1 mL titrant becomes appreciable. 

The equivalence point for this titration is indicated in Figure 14.1. Because the reaction is symmetrical, the equivalence point inflection point of the curve— that point at which it is steepest) occurs at the midpoint of the rising part of the curve. In nonsymmetrical titrations, the inflection point will not occur at the midpoint. For example, in the titration of Fe " with Mn04 , the steepest portion occurs near the top of the break because of the consumption of protons in the reaction, causing it to be nonsymmetiicaf. See Problem 21 for a spreadsheet calculation of the curve in Figure 14.1. 

An Excel spreadsheet can be constructed with appropriate formulas (to include the effects of dilution of the sample by titrant) to simulate the titration of weak and strong acids and bases (Figure 18.21). Some simulations use a master equation to calculate all points on the titration others use separate equations for different regions of the curve, for example before the equivalence point, at the equivalence point and after the equivalence point. The concentration of different species at a particular pH is calculated from [H (aq)l, and the volume of titrant required to produce that amount of each species is calculated. 

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