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Curve-Fitting Approach to Potentiometric Titration Using Spreadsheet Ngai Ling Ma and C. W. Tsang Department of Applied Biology and Chemical Technology, Hung Hom, Hong Kong Polytechnic University, Hong Kong Potentiometric titration is a fairly standard undergraduate chemistry experiment. A typical application of the potentiometric titration is to determine the solubility product of a sparingly soluble salt, such as silver chloride (1), from the titration endpoint. The endpoint can be determined by inspection of the titration curve (Fig. 1a), or more precisely, from the maximum of the first derivative or the x-intercept of the second derivative of potential with respect to titrant volume (Fig. 1b). Although most students have no problem with the theory, the analysis of the experimental data is tedious. Spreadsheet techniques have been introduced to facilitate the collection and analysis of data (2, 3). However, owing to the presence of experimental error, even when the moving-average method is applied to smooth the data, the result may not be satisfactory (Fig. 1b). Moreover, a slightly different endpoint may be obtained from the graphs because numerical differentiation is employed. Here, an alternative approach, using a curve-fitting technique, is introduced. While most students are familiar with fitting experimental data to a straight line, few realize that the same principle can be applied to other mathematical functions. A typical titration curve is shown in Figure 1a. Recognizing the similarity in the shapes of the titration curve and an arc tangent function, we designed an Excel 5.0 spread-
sheet template. In this template, students are first required to familiarize themselves with the shape of the arc tangent function given by eq 1:
Figure 1. (a) A potentiometric titration curve of KCl with AgNO3 . (b) The second derivative plot of the titration curve. Note that due to the presence of experimental error, oscillations are observed around the endpoint even if three-point average smoothing technique is applied.
Figure 2. (a) A potentiometric titration curve of KCl with AgNO3 using the same set of data as in Fig. 1. Note that the arctangent function is required to give a better fit in the steep region. (b) The second derivative plot of the titration curve. Since the potential is modeled as an analytic function, the second derivative plot is smooth.
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y = a + b * tan{1 [c (x – d) ]
(1)
In particular, they are required to take notes on the effect of varying the parameters a, b, c, and d on the shape and position of the curve. On another page of the template, the student enters the volume of titrant and measured potential and the experimental titration curve will be plotted automatically. Based on their observations on how each parameter in eq 1 affects the function, students are asked to devise an algorithm for fitting the model function to the experimental curve. With their designed algorithm, students carry out their fitting procedure modeled by eq 2 E = a + b * tan{1 [c (V – d) ]
(2)
where E and V are the measured potential and titrant volume, respectively. The progress of the curve-fitting process can be monitored by visual inspection of the difference between the experimental and the model function or by the sum of the root-mean-square (rms) error. It is important to note that eq 2 is only a model, and thus a perfect fit cannot be obtained. Given that the aim of the experiment is to de-
Journal of Chemical Education • Vol. 75 No. 1 January 1998 • JChemEd.chem.wisc.edu
Information • Textbooks • Media • Resources termine the titration endpoint, it is more important to obtain a good fit in the steep region of the titration curve (Fig. 2a). Once the fitting parameters have been decided, the volume at endpoint is simply given by the value of parameter d, as the first and second derivatives of eq 2 can be obtained easily using rules of differentiation as eqs 3 and 4 (Fig. 2b):
dE = dV
2
d E dV
= 2
bc
(3)
2
1+ c V –d
{2 b c 3 V – d 2
2
(4)
1+ c V –d
most by 0.5 mL. Of the two methods, students prefer the curve-fitting approach because it avoids the problem depicted in Fig. 1b so that once the fitting parameters have been determined, the endpoint can be determined unambiguously. This approach to curve fitting using a spreadsheet has some advantages over black-box curve fitting programs. By visually comparing the experimental data with the model function, students gain a better feel for how the shape of a mathematical function and error relate to changes in the parameters. Letting students experiment with the parameters first builds their confidence in using eq 1. The exercise of deriving a fitting algorithm encourages discussions among students and integrates what students have learned in mathematics and computing to solve chemistry problems. Acknowledgment
Both the traditional approach of data smoothing and this curve-fitting approach have been programmed as spreadsheet templates and used in the physical chemistry laboratory by our first-year students. The data-smoothing approach is easier to use because it does not require any additional input from the students other than the experimental data. On the other hand, this trial-and-error curve fitting using spreadsheet might look more messy. However, in practice, if the students have thought carefully on how to go about fitting the data, they can usually achieve rms error of less than 10 units within 10 to 15 trials. The endpoints determined from the two methods usually differ at
I wish to thank Annie F. M. Siu, who worked so diligently through her summer vacation, without pay, to help me revise various experiments in our first-year physical chemistry laboratory manual, including the potentiometric titration. Literature Cited 1. Matthews, G. P. Experimental Physical Chemistry; Oxford: New York, 1985. 2. Leharne, S.; Metcalfe, E. Ed. Chem. 1989, Sept, 143–147. 3. Mullin, J.; Marquardt, M. J. Chem. Educ. 1995, 72, 400–401.
JChemEd.chem.wisc.edu • Vol. 75 No. 1 January 1998 • Journal of Chemical Education
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