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### Keywords:

• Buffering;
• graphical analysis;
• mathematics education;
• science education;
• titration

### Abstract

We describe a simple undergraduate exercise involving the titration of a weak acid by a strong base using a pH meter and a micropipette. Students then use their data and carry out graphical analyses with a spreadsheet. The analyses involve using mathematical concepts such as first-derivative and semi-log plots and provide an opportunity for collaboration between biochemistry and mathematics instructors. By focusing on titration data, rather than the titration process, and using a variety of graphical transformations, we believe that students achieve a deeper understanding of the concept of buffering.

There is a growing emphasis on interdisciplinary approaches in science, mathematics, and technology education. A natural integration of subjects exists between biochemistry and mathematics. Biochemists routinely use mathematics and graphical methods to analyze data and to make sense of experimental information. Mathematics is also essential in helping students evaluate and clarify observed phenomena by modeling idealized situations, and such data have been successfully used to provide relevance in the mathematics classroom [1]. One topic in every introductory biochemistry course in which a synergy between mathematics and science is essential for student understanding is buffering.

Most students enter biochemistry courses having had some experience in titration of weak acids with a strong base from the general chemistry laboratory. However, the value of this prior experience is often limited with regard to an understanding of the concept of the ability of weak acids to buffer solution pH. What is emphasized is the value of the equivalence point in determining the amount of the weak acid present in the solution. In talking with students in the biochemistry teaching laboratory, their description of the details of titration is fuzzy, and the one general recollection from the prior titration exercise is that the indicator became colored at the end-point.

Having students conduct a titration and graphically analyze the data with a focus on exploring the concept of buffering, rather than the mechanics of the titration, can produce a deeper understanding of the process [2]. Such an exercise can also provide an opportunity to illustrate important concepts of mathematics that underpin chemistry and biochemistry. Over the past two decades, a number of reports have described the use of computer spreadsheets to provide educational models [3, 4] or to enhance titration exercises by providing students with more powerful analyses of the trends and error in the data [5, 6]. Other reports have focused on titration from a quantitative perspective and have included graphical analyses, such as the Gran Plot to let students clearly identify the equivalence point during the titration of monoprotic acids [79] or the Bjerrum Plot to estimate the pKa values of multiprotic acids [10].

For the graphing of titration data, it appears that there is no single convention for the plot to be used. Most biochemistry textbooks feature a plot with volume or equivalents of strong base on the x axis and pH on the y axis [1117], whereas at least two texts use pH on the x axis and the amount of strong base on the y axis [18, 19]. Similarly, articles in the science education literature present titration plots with amount of strong base on the x axis [5, 20] or pH on the x axis [21]. One interesting variation was to focus on “buffering capacity,” the plotting of the first derivative of the buffering capacity (Δml base/ΔpH) as a function of pH [21]. The concept of buffering capacity (originally called buffering value) was formalized by van Slyke in 1922 [22] and does not seem to have been formally presented in biochemistry textbooks since 1966 [23]. This plot produced a graph with a positive peak, making it obvious to students that maximal buffering occurred in the vicinity of the pKa of the weak acid employed in the titration. However, the above article [21] focusing on buffering capacity used computer-generated titration data, idealized to the extent that it even included theoretical back titration with HCl to fully protonate the weak acid. It was unclear if this method of graphical analysis would be pedagogically as useful using student-generated laboratory data.

In this article, our intent is to focus on how titration data can serve as a focus for explicitly integrating the mathematics concepts that students have learned into a laboratory exercise with increased emphasis on the concept of buffering. The mathematics topics we will employ include logarithms and derivatives, as well as an examination of the effect of axis choice in graphical analyses. The intent of emphasizing the synergy of mathematics and biochemistry is to facilitate student understanding of the concept of buffering, but we also find that providing a variety of ways of visualizing data accommodates students with multiple learning styles and is consistent with the use of cognitive load theory to facilitate encoding of concepts from working memory into permanent memory.

### BACKGROUND PROVIDED TO STUDENTS

The importance of preparing students before an exercise such as this cannot be overemphasized. Student prior knowledge is highly correlated with learning in all situations [24] and is a critical component of meaningful learning in a teaching laboratory environment [25]. We suggest that this exercise will produce optimal learning when an instructor reviews the concepts of the protonation and deprotonation of weak acids in response to added bases and acids and reminds students of prior titrations using a burette and pH indicator. The instructor should then explain the difference in the type of data obtained in doing an end-point titration with an indicator versus continuously monitoring solution pH with a pH meter. The intent of this exercise is to provide students with the obvious synergy between mathematics and science, and we encourage science instructors to develop specific handouts of instructions for their students in collaboration with a mathematics instructor. Such collaboration could provide the basis for formation of an interdisciplinary module as recently recommended by the National Research Council BIO2010 Report [26].

In the formulation of “buffering value” by van Slyke [22], the following equation is used.

• (1)

The symbol β represents the buffering value and B represents the moles of strong base added to the solution containing the weak acid. As strong base is added to a solution containing a protonated weak acid (HA), there is a progressive deprotonation to produce the conjugate base (A). The mole fraction of the conjugate base form (α) is expressed by the following equation.

• (2)

Using this nomenclature, the Henderson-Hasselbalch equation may be expressed in terms of the mole fraction of the conjugate base.

• (3)

Because α, the fraction deprotonated, is proportional to the number of moles of strong base added to a 1 M solution of weak acid, the buffering value of the solution can be expressed in these terms and will be proportional to the value of β, which is expressed in terms of moles/pH.

• (4)

This function will increase as the fraction deprotonated approaches the pKa and by substituting the value α = 0.5, students can calculate that it will peak at the maximum value of 0.575 for a 1 M solution of weak acid.

#### The Conventional Plot—

Students will probably have seen titration curves with amount of strong base (NaOH) on the x axis and pH on the y axis in previous courses and in their textbook. Such a plot for acetic acid is shown in Fig. 1A. It is our experience that many students have difficulty in viewing this curve to see that the pKa is represented by an inflection point and understanding buffering in terms of decreased pH change with added base in the vicinity of the pKa.

#### The First Derivative Plot—

The number of students who are able to understand buffering seems to increase when the data is plotted as a first derivative (ΔpH/Δml) versus the equivalents of NaOH (Fig. 1B). The first derivative of the curve in Fig. 1A should produce a negative peak with a minimum value at the inflection point of the titration curve. This curve clearly shows a decrease in the pH change per NaOH added toward the middle of the titration, but it will not be obvious to most students how this plot can help to estimate the pKa of an unknown weak acid. Students should also be informed that, unlike the idealized data in Fig. 1, their titration data will go past the end-point for the titration. They should expect that plotting data as ΔpH/Δml versus the volume of NaOH will accentuate the equivalence point, giving them a sharp peak when the titration is completed.

An additional observation for students on Fig. 1B is that it is often difficult to see features such as the actual minimum in the data points. As the data approach the pKa, the ΔpH/Δml gradually approach a minimum, but it is not readily visible because of the large range of the points causing compression of the points with lower values. Such uneven distribution of data is common in science and mathematics. One way to change the distribution of the data is to use a semi-log plot. Suggest that students should try transforming their first derivative plot as log (ΔpH/Δml) versus NaOH to see if it makes the region of maximum buffering more obvious.

#### Viewing Titration Data as a Function of pH—

The next sequence of graphs (Fig. 2) begins by reversing axes to make a plot that depicts the volume of NaOH as a function of pH. By reversing the axes, the graphs emphasize changes that occur as a function of pH during the titration. The advantage of depicting the volume of NaOH as a function of pH (Fig. 2A) is that it seems to more clearly reveal to students a region during the titration in which a relatively large amount of titrant was added with correspondingly small pH change. Many students find this view of the data makes it easier to understand the functional consequence of buffering action than the more usual depiction in Fig. 1A. Additionally, the horizontally flattened portion of the curve indicates that at the end-point very little NaOH is added with an attendant large change in pH. Even though the data is the same as used in Fig. 1, the presentation in this alternate format seems to increase the proportion of the students in the class who are able to understand and describe the difference between the pKa (or midpoint, or inflection point) and the equivalence point of a titration.

#### Plotting to Emphasize Volume Changes During Titration—

The first derivative (Fig. 2B) of the data plotted in Fig. 2A reveal a strong positive peak in the slope (Δml/ΔpH) in the region of the titration near the midpoint (pKa) and the end-point of the titration. The strong indicator of the pKa was not evident in the more usual plot of the data as a function of the volume of titrant (Fig 1B); the reciprocal of the decreasing values caused by buffering in Fig. 1B are now more clearly a definite positive peak with a clear maximal value. The antilog of the pKa value (about 4.7) deduced from Fig. 2B provides the student with a reasonable estimate of the Ka (about 2 × 10−5M) of acetic acid. It should be noted that the reason most educational sources do not plot pH values on the x axis as in Fig. 2A is because of the tradition of placing the independent variable (or cause), equivalents of NaOH in this case, on the abscissa to produce emphasis on causal influence for the dependent variable. However, when the data are transformed as in Fig. 2B, the positive peak in Δml/ΔpH as a function of pH creates an upward peak. The natural tendency of humans to perceive and process deviations from a horizon [27] seems to help students identify and understand the nature of buffering.

### THE LABORATORY EXERCISE

The students carrying out this exercise had experienced use of micropipettes and pH meter calibration in previous exercises in the course. The students were expected to be familiar with the written protocol for the exercise and were given a 20-min orientation before the start of the exercise that reviewed deprotonation of weak acids and the expected shape of a titration curve based on the Henderson-Hasselbalch equation. For the actual exercise, students worked in pairs. Students calibrated their pH meter, obtained 20 ml of the 1 M acetic acid solution that had been prepared by the teaching assistant, and prepared 50 ml of a 1 M solution of NaOH using a digital balance, a flask, and a graduated cylinder. For the titration, the NaOH titrant was added in 0.5-ml aliquots with a micropipette to a stirred beaker containing the acetic acid solution and the electrode. The new pH reading following addition of the NaOH was generally stable within 30 s of the addition and was recorded. For our students, with prior experience in using micropipettes and calibrating the pH meter, the total time to carry out the exercise was less than 45 min.

### POST-LABORATORY ANALYSES

Following completion of the exercise, students entered the data into a computer spreadsheet for further evaluation and manipulation. Students were provided with a Microsoft Excel spreadsheet that contained the necessary formulae to transform the data for graphing. A copy of this spreadsheet may be downloaded from www-class.unl.edu/biochem/buffers.xls. To prepare the composite figures used in this manuscript, the data were transferred to a graphing program; students can produce similar single graphs within most spreadsheet programs.

How essential is access to a computer laboratory in helping students use this exercise to understand buffering? It is pedagogically desirable to have students analyze the data immediately. They will be preparing the graphical analyses of the data while the details discussed in the orientation are still fresh in their minds. But perhaps equally important is that having the instructor present in the computer laboratory provides an opportunity for intrinsic feedback. Intrinsic feedback, sometimes called cognitive or performance-related feedback, is more likely to result in student learning than delayed feedback [28, 29]. If there is not access to a computer laboratory, an alternative might be to insist that students carry out the spreadsheet analyses within the next day and respond to a number of questions by E-mail or a discussion board. This will permit the instructor (or other students if using the discussion board) to assist students that seem to be having difficulty with the concepts. The least desirable mechanism to help students learn about buffering would seem to be the traditional laboratory report that is prepared a week after the exercise, requiring the student to then wait another week for delayed, extrinsic feedback on the report.

### INSTRUCTIONS FOR STUDENT ANALYSES

Within 24 h, you should provide the following plots of your titration data within an Excel spreadsheet to your teaching assistant via E-mail attachments (or courseware drop box).

• Fig. 1: A plot of ml NaOH (x axis) versus pH (y axis)

• Fig. 2: A plot of ml NaOH (x axis) versus ΔpH/Δml (y axis)

• Fig. 3: A plot of ml NaOH (x axis) versus log ΔpH/Δml (y axis)

• Fig. 4: A plot of pH (x axis) versus ml NaOH (y axis)

• Fig. 5: A plot of pH (x axis) versus Δml/ΔpH (y axis)

• Fig. 6: A plot of pH (x axis) versus log Δml/ΔpH (y axis)

On each plot be sure to insert the figure number. Also indicate in your E-mail text the volume of NaOH it took to reach the end-point; the teacher assistant will post these end-point values on the class web site. Your name and the exercise number should be in the title of the file (e.g. JSmithEx7.xls). On plot 5, use arrows and text to label the pKa and the end-point of the titration. Also point to the part of the graph that indicates buffering and insert a sentence about what buffering means.

For the next laboratory meeting, prepare a laboratory report following the usual format that presents your analysis of the titration that you carried out. Include copies of the above six figures and discuss what you can conclude about the titration from each. Be sure to label the pKa and end-point on plot 5 and write a sentence on the plot about what buffering means.

### SPECIFIC QUESTIONS

• Does the initial pH of your acetic acid solution indicate that the mole fraction of A was zero? If not, what was the mole fraction A?

• Estimate the accuracy with which you could determine the end-point of the titration and the pKa of acetic acid from plots 2 and 3.

• Estimate the accuracy with which you could determine the end-point of the titration and the pKa of acetic acid from plots 5 and 6.

• How was the use of the first derivative function useful in these analyses?

• Comment on the end-points determined by the different student groups; what might be possible sources of error?

• What is your best estimate for the pKa of acetic acid?

• What is your best estimate for the Ka of acetic acid?

• How does this compare with values in the literature for the pKa or Ka for acetic acid from three different references?

• Comment on the usefulness of using a log function to identify the pKa values in Figs. 3 and 6.

### STUDENT RESULTS

Fig. 3 represents student-generated data using the above protocol and plotted as a function of the NaOH volume added. It is clear from Fig. 3B that the end-point produces a sharp peak, but the decrease in ΔpH/ΔNaOH in the vicinity of the pKa value is not sufficient to accurately estimate the pKa value without reference to the volume of NaOH necessary to reach the end-point. Using a semi-log plot (Fig. 3C) appears to narrow the pH range that is likely to represent the pKa value. Plotting the student data as a function of the pH during the titration is seen in Fig. 4. The plot of ΔNaOH/ΔpH versus pH (Fig 4B) clearly reveals a strong peak due to buffering and permits a fairly certain determination of the pKa value for the acetic acid. It should be clear to students that this plot is not as useful in determining the end-point of the titration as Fig. 3B. Also, because the data in Fig. 4B were not as compressed as that in Fig. 3B, there was little improvement produced by plotting them on a semi-log plot (Fig. 4C).

### DISCUSSION

Understanding the concept of buffering is neither trivial nor intuitive for most students in introductory biochemistry courses. Although students have been exposed to titrations of weak acids in prior chemistry courses, the focus in these courses has generally been on measuring the amount of weak acid present in the solution rather than on the buffering caused by deprotonating the weak acid. Also, it is common in many biochemistry courses to focus on the skills of solving standard pH problems, rather than on a deep understanding of the ability of a weak acid to provide resistance to solution pH change.

We have observed from working in teaching laboratories that doing titrations tends to focus students on the mechanical aspects of the procedure, thereby encouraging them to treat it as a “cookbook” experience. Much time and energy can be spent learning the proper handling of a burette, how to read an end-point, and the use of a titration-thief. We have decided to alter the approach to titration so that the focus is on understanding the concept of buffering. Consequently, we do not use a burette, utilizing instead micropipettes and pH meters that students have used multiple times in prior undergraduate laboratory exercises. We do not use a primary standard to determine the exact concentration of the titrant, but rather rely on preparing the solution by weight from a newly purchased bottle of NaOH pellets. There seems to be a pedagogical advantage in using 1 M weak acid and titrant in that the end-point is reached when approximately equal volumes of each have been combined. Also, we do not use freshly boiled water to make the weak acid or titrant, but rather utilize sufficiently high concentrations that the amount of carbonate contamination will be small enough not to be visible in the data [30]. The focus of the exercise is on achieving a student understanding of the concept of buffering.

In the past, we have had students enter similar titration data into a spreadsheet containing functions based on the Henderson-Hasselbalch equation and had them input pKa values to generate a theoretical line superimposed on their data points. Students would then use an empirical trial and error approach to find a best fit to their data. While this exercise seemed to make students more aware that laboratory data often do not look as perfect as data depicted in textbooks, it did not seem to foster a deep understanding of the concept of buffering. The use of graphical analysis to determine the pKa value from a positive peak on a graph appears to be more “concrete” to many students than using equations to calculate a best fit to experimental data points. Plotting ΔNaOH/ΔpH versus pH helps students to visualize that more volume of titrant is necessary to change the solution pH near the pKa.

An additional device that we use to facilitate student understanding is to have students mark their graphs with arrows and text to indicate pKa, end-point, and the zone of buffering. By having students put meaningful text on graphics, we take advantage of the “multimedia effect” that, according to cognitive load theory [31], enhances encoding of elements in working memory into permanent memory.

Using exercises such as the one described herein, students can see “real world” applications of mathematics and the utility of math in science. Interdisciplinary cooperation can strengthen the science-math connections for students. Mathematics teachers may want to use data generated in the biochemistry teaching laboratory for mathematics students to analyze. For example, a titration curve between a weak acid and a strong base can be used to discuss the logistic function in mathematics class and the adoption of pH as a unit of acidity, rather than [H+]. Having students compare data tables as entered into spreadsheets and graphs of the same data can convince students of the value of graphical analyses. The first derivative of a titration curve can be used to facilitate identification of both the inflection point at the pKa value and the end-point in chemistry class. By data transformation, the significant relationships can be made more pronounced and perhaps more easily understood.

As seen in the analysis of data in Figs. 3C and 4C, having students use a semi-log plot permits them to visualize data trends within a set that differ by several orders of magnitude. The ability to visualize these data makes the point to students that the volume of titrant needed to produce pH change at the equivalence point is small indeed, and that is why it is so difficult to precisely determine the end-point during a titration. Plotting the data as Δml/ΔpH versus pH as in Fig. 4, B and C, conceptualizes how relatively large amounts of NaOH are used in the pH range in which the weak acid serves as a buffer and miniscule amounts are needed for equivalent pH changes at end-point. Given this information, students are better able to understand the common experience of overshooting the end-point during a titration. These graphs also reinforce the concept that the maximum amount of buffering capacity occurs within one pH unit of the pKa.

The deliberate use of mathematics concepts into the exercise creates an opportunity for collaboration between a chemist or biochemist with a colleague from the mathematics faculty. Such integration of mathematics into science is recommended by the National Research Council [26] at the undergraduate level. Adaptation of this exercise to the high school level would be consistent with the National Education Standards [32]. The National Council of Teachers of Mathematics also recommends use mathematical models to represent and understand quantitative relationships in systems outside of mathematics [33]. We feel that the graphical analysis of buffering during a simple titration described in this report is an ideal system for a mathematics-science collaboration and can lead to a deeper understanding of the fundamental concept of buffering by students in biochemistry laboratory courses.

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