Buffer Squares: A Graphical Approach for the Determination of Buffer

Article Cite This: J. Chem. Educ. XXXX, XXX, XXX−XXX

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Buffer Squares: A Graphical Approach for the Determination of Buffer pH Using Logarithmic Concentration Diagrams Spiros A. Pergantis,*,†,‡ Iakovos Saridakis,† Alexandros Lyratzakis,† Leonidas Mavroudakis,†,‡ and Tamsyn Montagnon† †

Department of Chemistry, University of Crete, Voutes Campus, Heraklion 70013, Greece Environmental Chemical Processes Laboratory, University of Crete, Voutes Campus, Heraklion 70013, Greece

J. Chem. Educ. Downloaded from pubs.acs.org by UNIV OF LOUISIANA AT LAFAYETTE on 04/03/19. For personal use only.

‡

S Supporting Information *

ABSTRACT: Teaching the concept of pH buffers is considered to be important both in the final high-school years and at the early undergraduate level. Here, we propose the use of pH−log C diagrams to investigate the properties of pH buffers. This graphical approach is extremely simple to employ because it only requires drawing a simple square that can then be used to determine relevant pH-buffer parameters. This square is based on the Henderson−Hasselbalch equation with the length of each of its sides equal to abs(pH − pKa) and abs(log Cb − log Ca). In addition, the “buffer square”, as we propose naming it, can be used by instructors as a pedagogical tool to introduce the concept of buffer capacity, to help determine pH change upon the addition of an acid or base, and to easily calculate the required concentrations for preparing a pH buffer with specific properties. Finally, we consider this approach to be especially powerful for helping students visualize the location of a buffer system on a full pH−log C diagram and, thus, help them evaluate if the Henderson−Hasselbalch equation is valid for accurate pH determination as an alternative to the more complex cubic or quadratic equations that are needed to describe acid−base equilibria more precisely in some cases. KEYWORDS: First-Year Undergraduate/General, Analytical Chemistry, Biochemistry, Environmental Chemistry, Analogies/Transfer, Mnemonics/Rote Learning, Acids/Bases, Aqueous Solution Chemistry, pH

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INTRODUCTION Diagrams of pH−logarithmic concentration (log C) have been used extensively to enhance conceptual understanding of acid− base equilibria and to help solve equilibrium problems of varying difficulty without the need to resort to complicated mathematical calculations. The pH−log C coordination system was first introduced by Niels Bjerrum, a Danish chemist, in 1915.1 Since then, numerous scientists have contributed towards making these diagrams more useful, popular,2,3 and universally applicable.4−6 Currently, only a limited number of chemistry textbooks introduce this approach to help students understand acid−base equilibria. Among these examples, a recent undergraduate analytical chemistry textbook has demonstrated the use of pH−log C diagrams in solving numerous acid−base-equilibrium problems.7 Prior to this publication, a tutorial book by Robert de Levie gave an extensive account on the use of pH−log C diagrams.8 The most comprehensive mathematical coverage of the pH−log C diagram topic was published recently as a book authored by Kahlert and Scholz.9 Educational papers on this topic published in scientific journals, including the J. Chem. Ed., have also appeared over the years, with the objective of promoting the use of pH−log C diagrams in undergraduate teaching.10−12 © XXXX American Chemical Society and Division of Chemical Education, Inc.

The main advantages of constructing and using pH−log C diagrams is that they provide detailed information about the concentrations of acid−base-equilibrium species over several orders of magnitude. This helps chemists to visualize the distribution of chemical species and, thus, allows them to decide which species are important and which can be ignored in order to conveniently calculate relevant chemical parameters. Another attractive feature is the fact that most associated problems (e.g., what the pH of a weak acid solution is) can be solved using the diagram without any requirement for timeconsuming and complicated mathematical calculations. In addition, these diagrams can be used to assist in the drawing of titration curves, a tedious task when done by using calculations alone.12 Constructing and using pH−log C diagrams, as described by several authors,7−11 is a relatively trivial task once a few basic rules have been understood and followed. Even though the approach has many benefits it also has some limitations. As described by Kovac,11 these limitations include their use in the treatment of buffer solutions. Because buffer systems are very close to the system point and, therefore, Received: July 22, 2018 Revised: March 17, 2019

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DOI: 10.1021/acs.jchemed.8b00588 J. Chem. Educ. XXXX, XXX, XXX−XXX

Journal of Chemical Education

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fall within the curved region of the pH−log C diagram, it is difficult to study them using hand-drawn diagrams. As a result, most of the existing approaches are not sufficiently accurate in this region. With this in mind Kovac concludes, “A carefully drawn diagram can be used to solve buffer problems, but it is probably better to use the Henderson−Hasselbalch equation and solve the problem algebraically.”11 In the present tutorial paper, we wish to demonstrate that indeed buffer systems can be conveniently studied using pH−log C diagrams. In this way relevant information regarding buffer pH can be extracted conveniently, along with the acid and conjugate-base concentrations needed to make a buffer with a specific pH, the pH changes observed when adding known amounts of strong acid or base to a buffer, the buffer’s capacity, and the parameters that affect it. Most importantly, the procedure proposed here does not include any “difficult to draw” figures but uses only the simple drawing of four intersecting straight lines to form a square. In fact, the student does not need to have any prior knowledge of drawing and manipulating conventional pH−log C diagrams to study the full range of an acid−base equilibrium. However, some basic knowledge of the topic may help the reader better understand and appreciate the benefits of using pH−log C diagrams to study buffer systems. We must stress that the proposed buffer-square approach is not intended to replace the way in which pH buffers are currently being introduced in general-chemistry courses. It is rather intended as a complementary approach to help students better understand pH-buffer concepts and how to graphically solve pH-buffer problems. Finally, it also provides a convenient and difficult-to-forget mnemonic for remembering the Henderson−Hasselbalch equation.

Figure 1. pH−log C diagram showing a square (ABCD) that represents the Henderson−Hasselbalch equation describing a pH buffer. We have named this square the buffer square (b-SQ), whose sides cannot be greater than 1 log unit, and thus it cannot have an area greater than 1, or else the system is not considered a pH buffer, because its buffering capacity drops off significantly outside these limits.

Therefore, the b-SQ can have a maximum side length of 1 log unit. As a result, if the b-SQ has an area greater than 1, then the system under investigation is not an efficient pH buffer. In this case, the b-SQ graphical approach is not suitable, and the full pH−log C diagram approach becomes more relevant for determining the system’s pH.

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HOW TO CONSTRUCT A pH−LOG C DIAGRAM TO DETERMINE BUFFER pH In order to examine pH-buffer properties in a graphical fashion, we can apply any one of the following three graphical approaches. These approaches include (a) using a blank piece of paper to hand-draw the b-SQ and obtain relevant parameter values, (b) using graph paper to construct the b-SQ and extract relevant pH-buffer parameters, and (c) using a spreadsheet to construct the b-SQ and extract relevant parameters. All three approaches are simple to implement, and so the approach chosen ultimately depends on what is available to the student or instructor (blank paper, graph paper, or software) and what best matches the learning objectives set by the instructor.

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THEORETICAL BACKGROUND OF THE GRAPHICAL APPROACH FOR BUFFER-pH DETERMINATION The graphical approach proposed in this tutorial for the determination of buffer pH is based on the Henderson− Hasselbalch equation following the rearrangements shown below: pH = pK a + log

Cb ⇒ Ca

Using a Blank Piece of Paper to Construct a pH−log C Diagram for Determining Buffer pH

pH = pK a + (log C b − log Ca) ⇒ (pH − pK a) = (log C b − log Ca) ⇒ abs(pH − pK a) = abs(log C b − log Ca)

Let us consider a pH buffer containing CH3COOH (Ca = 0.020 M and pKa = 4.77) and CH3COONa (Cb = 0.050 M). How do we determine its pH using a graphical approach on a blank piece of paper? First of all, the log Ca and log Cb values need to be calculated, as is always necessary when constructing pH−log C diagrams. Thus, log Ca = log(0.020) = −1.7 and log Cb = log(0.050) = −1.3. The initial step in this graphical procedure is to draw an ABCD square anywhere on an imaginary xy-coordinate system (x = pH, y = log C), with point A being the top left corner and all subsequent points following in a counter-clockwise fashion (Figure 2). The size of the square does not matter as long as it is a square. Subsequently, the sides of the b-SQ can be labeled according to the parameters of eq 1; the opposite vertical sides, AB and CD, correspond to the pKa and pH lines, whereas the opposite horizontal sides, BC and AD, correspond to the log Ca and log Cb lines. The left vertical line, AB, corresponds to the pKa or

(1)

Equation 1 can be represented graphically on a pH−log C diagram (the x-axis is the pH axis, whereas the y-axis is the log C axis) by drawing a geometric figure having a vertical side with a length equal to log C b − log Ca and a perpendicular horizontal side with a length equal to pH − pKa. From eq 1, it is apparent that the two sides have equal length. These two sides form the basis for drawing a square in the pH−log C diagram (Figure 1). In the remainder of this manuscript, we will refer to this Henderson−Hasselbalch-derived square as the “buffer square” (b-SQ). Throughout this tutorial, the b-SQ will be used to determine several pH-buffer parameters. However, it can only be used within the range of 0.1 ≤ Cb/Ca ≤ 10, where an acceptable buffer capacity can be achieved (i.e., −1 ≤ log C b − log Ca ≤ +1 and, thus, −1 ≤ pH − pKa ≤ +1). B



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buffer parameters (Figure 2). Because the distance between the AD and BC lines equals the absolute difference between log Cb and log Ca (=0.4), and we are dealing with a square, the difference between the pKa and pH must also be 0.4. Thus, the pH is 0.4 greater than the pKa value (4.77); that is, pH = 4.77 + 0.4 = 5.17. Note that the plus sign (+) used here would be minus (−) if the pH line was to the left of the pKa line (i.e., log Cb < log Ca). A logical mnemonic for conveniently constructing the buffer square is that when Ca > Cb, its sides should be in the following order when starting from side AB and going clockwise around the square: pH line (side AB) → log Ca line (side AD) → pKa line (side CD) → log Cb line (side BC). In contrast, if we know that Cb > Ca, then the order of the square sides, starting from side AB and going clockwise, should be pKa line (side AB) → log Cb line (side AD) → pH line (side CD) → log Ca line (side BC). Using Graph Paper to Construct a pH−log C Diagram for Determining Buffer pH

To demonstrate the application of the slightly more detailed graph-paper approach for determining buffer pH, an NH3/ NH4Cl buffer system with the following properties is considered: NH4+ with Ca = 0.040 M and pKa = 9.26 and NH3 with Cb = 0.060 M. On the graph paper, the pH scale is represented on the x-axis, spanning a range from pKa − 1.0 = 8.26 to pKa + 1.0 = 10.26 (Figure 3). It is recommended that the midpoint on the pH axis should be a value close to the acid pKa. The range, however, may be selected in order to maximize resolution and, thus, achieve better accuracy for pH determination. The diagram’s y-axis, which corresponds to log C, can be drawn to have a range that depends on the log Ca and log Cb values and the difference between them. In the example discussed here, it has a range of 1 log unit. In our example, we have Ca = [NH4+] = 0.040 M and Cb = [NH3] = 0.060 M and, thus, log Ca = −1.40 and log Cb = −1.22, respectively. (Note that the log C calculations need to be made, which is also the case for all pH−log C diagrams.) Once the pH−log C diagram axes have been prepared, the following three lines need to be added to the diagram. First, the acid pKa is noted on the x = pH axis, and a vertical line is drawn (yellow line in Figure 3); subsequently, the log Ca and

Figure 2. Hand-drawn or back-of-the-envelope b-SQ suitable for determining the buffer-solution pH.

pH, whichever has the lowest value, and the right vertical line, CD, corresponds to whichever has the highest value. However, in the present example, this does not help in assigning each of the vertical sides, because the pH is unknown. However, because log Cb > log Ca (more base in the buffer than acid), the buffer pH must be greater than the acid pKa, and thus the pH will correspond to the right vertical CD line, whereas the pKa will correspond to the left vertical AB line. The opposite holds true when log Cb < log Ca. We can define the horizontal sides, AD and BC, as corresponding to the log Cb and log Ca lines, respectively. This is because the AD line (i.e., the top line) corresponds to the highest log C value, which is the log Cb value, whereas the bottom BC line corresponds to the component with the lowest concentration (i.e., log Ca). The constructed b-SQ has now been labeled with all the important

Figure 3. Buffer square (b-SQ) drawn within a pH−log C diagram and its use for determining buffer pH. C



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Figure 4. Snapshot of a pH−log C diagram drawn from a spreadsheet for buffer-pH determination. The red line (pH line) cuts the x-axis at pH = 7.37. Notice that the b-SQ is formed from the four lines corresponding to system’s pKa, log Ca, log Cb, and pH values.

log Cb points are located on the y = log C axis, and horizontal lines are drawn parallel to the pH axis (green log Ca line and blue log Cb line in Figure 3). These lines intersect the pKa line (yellow) at points B and A, respectively. The distance between points A and B can be easily determined visually on the graph paper by counting the squares between the two points. The distance AB is then added or subtracted horizontally to the pKa line at point B, resulting in a new point, C. Choosing to add or subtract depends on whether the buffer’s component base or acid has the highest concentration, respectively. For the buffer system given here, we can conveniently draw the pH line by starting from the pKa line (point B) and adding distance AB horizontally to its right. In the current example, the conjugate base has a higher concentration than the acid; therefore, the distance is added to point B, resulting in a new point, C, located to the right of B. From point C, a vertical line is drawn that intersects the pH axis to give the buffer pH. Using this approach, the buffer pH has been determined graphically to be 9.44 (Figure 3). It should be noted, that the pH line also cuts the log Cb line at point D, thus completing the b-SQ. Provided that the recommended graph ranges have been used, the precision of this approach is to the second decimal point of the pH value. In this way, no precision has been lost compared with the experimental precision and relative to the pH value derived from mathematical calculations. Overall, this is an extremely simple approach to utilize in order to determine the pH of a buffer, because it only requires a piece of graph paper and the ability to draw a square along with knowledge of the weak-acid pKa, the acid log Ca, and the conjugate-base log Cb values.

Ultimately, any of the b-SQ approaches described here can be used to graphically determine any of the four parameters shown in eq 1, provided that the distance between any two opposite sides is known, and the absolute value of a third side is also known. Using the b-SQ Approach to Better Understand Buffer Capacity

The b-SQ approach may also be used to provide students with better insight into a system’s pH-buffer capacity. If the b-SQ’s area is less than 1, because the square sides must be ≤1 log unit each, then we are dealing with a pH buffer; whereas, if the square’s area is greater than 1, its buffering capacity may be significantly reduced. Furthermore, because the b-SQ approach is based on the Henderson−Hasselbalch equation it may not be as accurate in this case, and therefore, a full pH−log C diagram should be used to determine pH and species concentrations. The reason the b-SQ area should be less than 1 is because it is generally accepted that in order for a buffer to have adequate buffering capacity, the acid−conjugatebase concentrations must be within the range 0.1 ≤ Cb/Ca ≤ 10 ⇒ −1 ≤ log Cb − log Ca ≤ +1; thus, −1 ≤ pH − pKa ≤ +1. Therefore, the b-SQ system becomes a simple tool for evaluating buffer capacity because high buffer capacity is associated with small squares having an area ≪1, low buffer capacity is associated with the b-SQ having an area close to 1, and maximum capacity is provided when the square has collapsed into a single point (the area is 0). Let us now further examine what the b-SQ can reveal about buffer capacity by assuming we have a 1.0 L buffer solution consisting of CH3COOH (Ca = 0.0016 M ⇒ log Ca = −2.80, pKa = 4.77) and CH3COONa (Cb = 0.0024 M ⇒ log Cb = −2.62). To apply the b-SQ approach, we draw a vertical pKa line at x = 4.77 and two horizontal lines at −2.80 and −2.62 (Figure 5a). The two horizontal lines cut the vertical pKa line at points A and B. The distance AB is then added to the vertical pKa line at point B to produce a new point C in the horizontal direction. Subsequently, a vertical line drawn from point C will give us the system’s pH, which is 4.95. Now, let us examine what will happen to the buffer pH if 0.30 mmol of H+ (monoprotic strong acid) are added (assuming no change in volume). The outcome of course has to do with the buffer’s

Using a Spreadsheet Program to Construct a pH−log C Diagram for Determining Buffer pH

To further increase the versatility of this approach, we have configured a macro-containing spreadsheet in order to automatically construct these lines in the way we have just described and, thus, provide immediate graphical determination of the buffer pH upon entering the values of the pKa, log Ca, and log Cb. A snapshot of the spreadsheet graph is shown in Figure 4. The actual spreadsheet file is provided as part of the paper’s Supporting Information. D



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Figure 5. Demonstrating buffer-pH capacity using the b-SQ approach before (a), during (b), and after (c) the addition of a monoprotic acid.

The vertical line we draw through point C′ cuts the x-axis at 4.81, which is the buffer’s new pH. This process clearly shows the high buffer capacity of the system under examination, because only a relatively small pH change occurred (pH 4.95 to 4.81), following the addition of a significant amount of acid to the system. If a similar experiment were to be conducted for the same acid−conjugate-base system but with the component concentrations outside the buffering area (i.e., the system having been adjusted to have a pH of 7), then a conventional full-scale pH−log C diagram must be used to determine the new pH (Figure 6). In this case, however, when the acid is added at

capacity to mitigate pH change upon the addition of acid or base. In the present example, the 0.30 mmol of H+ added will react with the conjugate base, and, thus, decrease its amount by 0.30 mmol/L, concomitantly affording an additional 0.30 mmol/L acid. Therefore, the new log Cb′ = log(0.0024 − 0.00030) = −2.68, and a new horizontal log Cb′ line should be drawn, as shown in Figure 5b. The acid concentration also changes, as log Ca′ = log(0.0016 + 0.00030) = −2.72. This is represented by another horizontal line, also shown in Figure 5b. These two new log C lines dissect the pKa line at points A′ and B′, respectively. Once again, the distance A′B′ is added to point B′ to produce C′ in the horizontal direction (Figure 5c). E



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However, this may not always be the case. For example, let us examine an acid−conjugate-base system with pKa = 2.26, Ca = 0.4 M, and Cb = 0.04 M. In this case, [H3O+] is about the same as Cb, and thus the former term cannot be ignored in eq 3. As a result, use of the Henderson−Hasselbalch equation is not very accurate, as it will give a pH of 1.26 instead of the more accurate pH 1.52 calculated using a version of eq 3 in which the term Kw/[H3O+] is omitted as negligible. This is a confusing point for students as they will sometimes fail to take this parameter into account and proceed with the use of the simplified Henderson−Hasselbalch equation. However, when using the b-SQ approach this limitation is more easily identified, especially when using the graph-paper approach to draw the b-SQ. All the student has to do is look for the [H3O+] or [OH−] lines close to or within the b-SQ that they have drawn (Figure 7a). This is simple because the [H3O+] line, which has a slope of −1 (the equation is y = −x, where y is log

Figure 6. Full-range pH−log C diagram for a solution prepared to contain CH3COOH/CH3COONa, adjusted to have pH = 7. Addition of 0.3 mmol/L acid results in the solution pH being reduced to 5.9. The blue box is the area where the buffering capacity of the system is at its maximum. Outside this box it drops significantly. The dotted red line shows the solution pH before the acid addition, whereas the solid red line shows the pH following acid addition. The solid blue line represents the CH3COOH species, whereas the dashed pink line represents the CH3COO− species. The diagonal solid black line represents the H+ concentration, which has a slope of −1, whereas the black dotted line represents the OH− concentration, which has a slope of +1.

0.30 mmol/L, the pH changes from 7.0 to 5.9. This dramatic pH change is due to the lack of buffering capacity when the acid−conjugate-base pair CH3 COOH/CH3 COONa are present in a solution with pH = 7.

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USING THE b-SQ APPROACH TO READILY IDENTIFY SITUATIONS FOR WHICH THE HENDERSON−HASSELBALCH EQUATION IS NOT VALID The exact cubic equation that describes ionic equilibria for a mixture containing a weak acid (HA) and its salt (NaA) is [H3O+]3 + (C b + K a)[H3O+]2 − (K w + K aCa)[H3O+] − K wK a = 0

(2)

This equation was derived by rearranging the following aciddissociation-equilibrium equation: [H3O+]{C b + ([H3O+] − K w /[H3O+])} = Ka Ca − ([H3O+] − K w /[H3O+])

(3)

To solve buffer problems, we assume that the amounts of [H3O+] and Kw/[H3O+] are usually significantly lower than Ca and Cb, and therefore, eq 3 can be simplified to give eq 4, which is readily converted into the Henderson−Hasselbalch equation (eq 1): [H3O+]C b ≈ Ka Ca

Figure 7. Buffer square for an acid−base system with the following parameters: pKa = 2.26, Ca = 0.4 M, and Cb = 0.04 M. (a) Close-up of the b-SQ (blue area). Note the [H3O+] line passing through the bSQ. (b) Full pH−log C diagram showing the b-SQ (blue area), with the solid blue line representing the nondissociated acid species (HA), and the dashed pink line representing the dissociated species (A−). The diagonal solid black line represents the concentration of H+ and has a slope of −1, whereas the black dotted line represents the concentration of OH− and has a slope of +1.

(4)

This assumption is usually valid because for the preparation of buffered solutions, Ca and Cb are usually maintained at high concentrations, generally between 1 and 0.02 M, in order to provide adequate buffer capacity. Thus, the Henderson− Hasselbalch equation can be used without a problem. F



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C, and x is pH), has the following (x, y) coordinate pattern: (0, 0), (1, −1), ..., (2, −2), ..., (7, −7), ..., (14, −14). So, if the student locates any such points on the b-SQ graph, there is cause for concern, because it means that a relatively significant H3O+ concentration from water self-dissociation is present, and thus more complex equations must be employed. This is also the case when the [OH−]-water-self-dissociation line passes close to or through the b-SQ. This can easily be checked because the [OH−] line has a slope of +1 (the equation here is y = x − 14 or log C = pH − 14) and thus the (x, y) coordinates of (0, −14), ..., (1, 1 − 14), ..., (2, 2 − 14), (3, 3 − 14), ..., (7, 7 − 14), ..., (14, 14 − 14). Once again, if any such points exist near or within the b-SQ, then using the Henderson− Hasselbalch equation may be problematic because water selfdissociation, which is now significant, has not been taken into account. All this analysis, of course, can also be visualized on the full pH−log C diagram (Figure 7b).

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HOW TO USE THE b-SQ APPROACH TO PLAN THE PREPARATION OF A BUFFER SOLUTION Let us assume that we want to prepare a buffer system consisting of CH3COOH (pKa = 4.77) and CH3COONa with a pH of 3.95. How can the b-SQ approach be used to help us prepare it? To examine this question, we need to draw a b-SQ as already demonstrated. To do this, we make the two vertical pH and pKa lines at 3.95 and 4.77, respectively (i.e., 0.82 log units apart; Figure 8). This means that log Cb − log Ca must also be equal to 0.82 log units. Thus, the resulting b-SQ is 0.82 log units by 0.82 log units. To find the acid−base concentrations that are required to achieve a pH of 3.95, we can move this b-SQ up and down the pH and pKa lines (i.e., we can treat these lines as “railway tracks”). Once we stop moving the b-SQ, we can extend its two horizontal lines to cut the y-axis at their corresponding log C a and log C b concentrations. The higher the b-SQ is on the “railway track” the higher the buffer capacity will be (i.e., higher concentrations of acid and base). Once these two concentrations have been decided upon, it is easy to calculate Ca = antilog Ca and Cb = antilog Cb. Their sum, Ca + Cb, can be the starting concentration of the acid CH3COOH used to prepare the buffer. Subsequently, an appropriate amount of strong base (NaOH) is added in order to partially convert the acid to its conjugate-base form (CH3COO−) and, thus, achieve a buffer with a pH of 3.95 as described by the b-SQ approach. The lower the b-SQ is located on the “railway tracks” (i.e., the more dilute it is), the lower its buffer capacity becomes. Finally, when it gets close to the diagonal [H+] and [OH−] lines, the Henderson−Hasselbalch equation becomes less accurate for determining the system’s pH, and thus the b-SQ approach is also less accurate.

Figure 8. Using the b-SQ approach to determine acid−base concentrations required to prepare a pH buffer with a specific pH.

Even though there is extensive educational literature describing pH buffers, it is based almost exclusively on the Henderson−Hasselbalch equation and its mathematical solution, with very few exceptions, such as the recently reported 3-D Surface Visualization of pH Titration “Topos” approach, which describes buffer plateaus.13 This means that students either remember the Henderson−Hasselbalch equation by heart and use it, hopefully correctly, or do not remember it and either have to derive it from the equilibriumconstant expression by ignoring species that are of negligible concentration or are simply unable to solve pH-buffer-related problems. Offering a unique graphical approach involving a simple “square”, a geometric figure the characteristics of which all students are well acquainted with from their earliest mathematical forays in primary school, provides students with an extremely user-friendly alternative way of solving pH-buffer problems. Instead of having to remember an abstract equation, they now only need to remember that a square consisting of a vertical abs(log Cb − log Ca) side and a horizontal abs(pH − pKa) side can be used to describe pH-buffer properties. Knowing the length of one side allows them to draw the other

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CONCLUSIONS In this tutorial, we have described a unique graphical approach that we hope will allow students to improve their understanding of pH buffers and their parameters. The buffer square is easy to construct as it is based on a square drawn in a pH− log C coordinate system. This square can be conveniently used to determine buffer pH if Ca and Cb are known, to determine the pKa of a buffer acid−conjugate-base pair when their concentrations and the solution pH are known, and also to determine the Ca and Cb required to make a buffer with a specific pH. G



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side as well. It also becomes easy for students to associate component properties with buffer capacity by remembering that smaller squares have greater buffer capacity than larger ones. Finally, squares with an area greater than 1 unit start to lose their buffer capacity significantly. At the very least, the pHbuffer-square approach can provide students with a way to check the validity of their Henderson−Hasselbalch-equation calculations. This new way to examine pH buffers and their properties may significantly help visually inclined students and should thus be taught to undergraduate students as it will probably be nigh on impossible for them to forget it. In some cases, it can even serve as the starting point to derive the Henderson−Hasselbalch equation if one has forgotten it. In this tutorial paper, we certainly do not claim that the buffer square approach is simpler or faster to use than the Henderson−Hasselbalch equation for calculating buffer pH. For the ammonia/ammonium ion buffer solution example described previously, all a student needs to do is drop three numbers into the Henderson−Hasselbalch equation and immediately calculate the resulting pH value. This is extremely simple; however, if he or she does not remember the exact form of the equation, things get a little more complicated. The student would have to spend time deriving the equation from acid−base-equilibrium theory, which they must know very well. In contrast, the buffer-square approach (in its simplest form using a blank piece of paper) only requires that the student draw a square and assign its sides in a logical fashion to buffer parameters (i.e., the vertical pKa line is parallel to the pH line, and the horizontal log Ca line is parallel to the log Cb line). The line order follows a logical sequence; if log Cb > log Ca then the pH line is right to the pKa line and vice versa. The labeled square that results contains information about all buffer parameters, so the b-SQ approach may in some cases be the faster and simpler approach.

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REFERENCES

(1) Bjerrum, N. Sammlung chemisch und chemisch-technischer. Vorträge 1915, 21, 1 (cited in Kahlert, H.; Scholtz, F. Acid-Base Diagrams; Springer-Verlag: Berlin, 2013). (2) Arnfeld, H. Svensk Kemisk Tidskrift 1937, 49, 96 (cited in Kahlert, H.; Scholtz, F. Acid-Base Diagrams; Springer-Verlag: Berlin, 2013). (3) Ö lander, A. Kompendium i teoretisk kemi; Tekniska Högskolan Stockholm: Stockholm, 1939 (cited in Kahlert, H.; Scholtz, F. AcidBase Diagrams; Springer-Verlag: Berlin, 2013). (4) Hägg, G. Die theoretischen Grundlagen der analytischen Chemie; Birkhäuser: Basel, 1950 (cited in Kahlert, H.; Scholtz, F. Acid-Base Diagrams; Springer-Verlag: Berlin, 2013). (5) Sillén, L. G.; Lange, P. W.; Gabrielson, C. O. Problems in physical chemistry; Prentice Hall: New York, 1952 (cited in Kahlert, H.; Scholtz, F. Acid-Base Diagrams; Springer-Verlag: Berlin, 2013). (6) Sillén, L. G. Graphical presentation of equilibrium data. In Treatise on Analytical Chemistry; Kolthoff, I. M., Elving, P. J., Sandell, E. B., Eds.; The Interscience Encyclopedia: New York, 1959. (7) Enke, C. G. The Art and Science of Chemical Analysis; John Wiley& Sons, Inc.: Hoboken, NJ, 2001. (8) de Levie, R. Aqueous Acid-Base Equilibria and Titrations; Oxford Press Primers; Oxford University Press: Oxford, 2006. (9) Kahlert, H.; Scholz, F. Acid-Base Diagrams; Springer-Verlag: Berlin, 2013. (10) Salvatore, F.; Salvatore, M. M. An Attractive Way of Developing the Concept of Systematic Titration Error of Visual Acid-Base Titrations (on the Basis of Logarithmic Acid-Base Diagrams). World J. Chem. Educ. 2014, 2 (1), 8−20. (11) Kovac, J. Using the Logarithmic Concentration Diagram, Log C, To Teach Acid-Base Equilibrium. J. Chem. Educ. 2012, 89 (7), 905−909. (12) Tabbutt, F. D. Titration Curves from Logarithmic Concentration Diagrams. J. Chem. Educ. 1966, 43 (5), 245−249. (13) Smith, G. C.; Hossain, M. M.; MacCarthy, P. 3-D Surface Visualization of pH Titration “Topos”: Equivalence Point Cliffs, Dilution Ramps, and Buffer Plateaus. J. Chem. Educ. 2014, 91, 225− 231.

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available on the ACS Publications website at DOI: 10.1021/acs.jchemed.8b00588. Calculations for pH change upon the addition of a monoprotic acid for Figure 6 (PDF) Macro-containing spreadsheet for automatically constructing buffer-square lines that provide immediate graphical determination of the buffer pH upon entering the pKa, log Ca, and log Cb (ZIP)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Spiros A. Pergantis: 0000-0002-9077-7870 Notes

The authors declare no competing financial interest.

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DEDICATION S.A.P. would like to dedicate this educational paper to Nikolaos P. Evmirides, Emeritus Professor at the University of Ioannina (UoI), Greece, who first introduced the topic of log C diagrams to him when he was an undergraduate student at UoI in the mid-1980’s. H


Buffer Squares: A Graphical Approach for the Determination of Buffer

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