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1 Buffer Capacity in Chemical Equilibrium How long can you hyperventilate before severe alkalosis sets in? Your blood pH is controlled (buffered) within a narrow range. In this module we will investigate the chemical characteristics of this buffer and estimate its physiologic limitations. This module will improve your understanding of buffer capacity through graphical analysis of the Henderson-Hasselbalch equation. Prepared for SSAC by Dr. Armando Herbelin – Lower Columbia College © The Washington Center for Improving the Quality of Undergraduate Education. All rights reserved. 2007 SSAC2007.QD561.AH1.1 – QD561 Core Quantitative Issue Algebra: Manipulating logarithmic equations Supporting Quantitative Issues Algebra: Manipulating equations, Solving simultaneous equations Modeling: What-if analysis Number sense: Dimensional analysis, Logarithms, Scientific notation Visual display of data: Rate of change, Slope, XY scatter plots

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2 Slide 3 Problem Specification Slides 4-7 Buffer Review (Includes three worked examples) Slide 8 Chemistry of the Blood Buffer System Slides 9-10 Graphing the Blood Buffer System Slides 11 Updated spreadsheet for disturbing the buffer system. Slide 12 Definition and Meaning of Buffer Capacity Slides 13-14 How much CO 2 do you lose per breath? Slide 15 End of Module Questions Overview of Module A chemical buffers ability to resist pH change, the Buffer Capacity, is important to the buffers practical performance. Calculation of buffer capacity is conceptually straightforward but involves repeated calculations. Application of a spreadsheet to repeat these calculations automatically simplifies the process and facilitates deeper understanding of its implications. After briefly reviewing several buffer calculations you will implement a spreadsheet to graphically investigate buffer capacity and characterize a physiological buffer – blood.

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3 Problem When someone hyperventilates, why do you have them breathe into a paper bag? Should you really be worried? Your blood pH is controlled between a pH of 7.35 and 7.45 by the bicarbonate/carbon dioxide (HCO 3 / CO 2 ) buffer system. When CO 2 dissolves in your blood, it combines with a water molecule to form carbonic acid, H 2 CO 3 : Carbonic acid is a weak acid that forms HCO 3 : Biologically, we focus on CO 2 and HCO 3 -. The equilibrium expression for a combined reaction is: When you hyperventilate, the CO 2 concentration in your blood drops. According to the Henderson- Hasselbalch equation, the pH will increase. Death occurs when blood pH reaches about 7.8. By quantifying the pH change, we can gaugehow serious hyperventilation might be. Question: How long can you hyperventilate before your blood pH reaches a dangerous level? Picture: www.uah.edu/ASEF/teachers.htm

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4 Buffer Review: Terminology Square brackets are used as a shorthand for concentration in molarity. –[H + ] = molarity of the hydrogen ion. pK a is the p of the acidity constant K a. p means negative log: pH = log[H + ] pK a = log(K a ) Use the following shorthand when you have to do algebra with chemical symbols: –HA = protonated (acid) form of the buffer molecule. –A = conjugate base form of the buffer molecule. –For example, acetic acid would be HA=CH 3 COOH, A = CH 3 COO Click here for an optional derivation of the Henderson- Hasselbalch equation.

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5 Buffer Review: pH from [HA] & [A - ]. Determine the pH of an acetic acid buffer formed from 0.025M acetic acid and 0.012M sodium acetate (K a =1.8 10 5 ) Givens: [HA] = 0.025M [A - ] = 0.012M pK a = log(1.8 10 5 )=4.74 To solve, plug into the Henderson- Hasselbalch equation.

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6 Buffer Review: [HA] and [A - ] for a given pH. What is the concentration of acetic acid and sodium acetate in a solution at pH=5.00? The total amount of buffer molecules is 0.037M. Given: –pH=5.0 –pK a = 4.74 (previous page) Total amount of buffer molecules is 0.037M means: –[HA]+[A ]=0.037M Solution: –Plug in pH and pK a, Solve the Henderson-Hasselbalch equation for the ratio of [A ]/[HA] –Solve two equations in 2 unknowns (A hassle) for [HA] and [A ] 2 Equations in 2 Unknowns Steps with red borders will be used in the spreadsheet calculations.

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7 Buffer Review: pH of a Disturbed Buffer. What is the pH of the previous buffer if 0.007M of acid is added. These problems are easier than they seem. Solution: –If acid has been added, if will convert conjugate base (A ) into the acid (HA). –[HA] goes up by 0.007M –[A ] goes down by 0.007M –Plug into the Henderson- Hasselbalch equation. Note: If these examples cause you to breathe into a paper bag, go back and review material on buffers in your chemistry textbook.

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8 The Blood Buffer System pH of Blood –Normal pH is approximately 7.35-7.45. –pH values compatible with life in mammals are limited to a pH range between 6.8 and 7.8. The blood is buffered by a carbonate buffer system with a total blood carbonate level of approximately 0.0048 M. –To make the numbers easier to work with, physiologists typically use millimolar as a unit (1mM = 0.001M) To work with buffers, we use the Henderson-Hasselbalch equation: The total blood carbonate is the sum of [HCO 3 ] and [CO 2 ] 6.0 6.57.0 7.5 8.0 AcidosisAlkalosisDeath Normal Click here for extra help with Excel before you start the buffers.

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9 Graphing the Carbonate Buffer System To help understand the blood buffer system, we will generate a spreadsheet to repeat the buffer calculations from Slide 6. Yellow cells contain a number. Orange cells contain a formula. To create your spreadsheet, start with your name and the date in Rows 2 & 3. Be sure to label it. Rows 4 & 5 have the constants for this problem – total buffer concentration and pK a. Column C has the pH values from 4.00 to 10.00 in increments of 0.3. Columns D-G are single steps through the calculation from Slide 6. Enter formulas for the calculations in Row 7 then copy and paste the cells for the whole column. Hint: Columns D and F should have a fixed references (pK a is in cell $E$5) Excel can automate the process of entering these numbers: 1.Enter 4.00 in Cell C7. Format the cell with 3 decimal places using the buttons. 2.Re-select Cell C7. 3.Open the Edit menu, Select Fill, and then Series. 4.Fill in the form as shown. Click Ok. 5.Now try this technique to fill in the numbers in Column B. Take a minute to read the spreadsheet then click to continue.

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10 Graphing the Carbonate Buffer System We will now generate a graph of the buffer system. Chart Wizard Step 1: Select XY Scatter, do not connect the dots. Click Next. Open the Insert menu and select Chart. Chart Wizard Step 2 (a): Select all text in the Data Range box and delete it. Chart Wizard Step 2 (b): Click the Series Tab. 1 2 3 4 Chart Wizard Step 2 (c): Click Add. Name: Click in the Name box then click on Cell C6 in your spreadsheet. X Values: Click in the X Values box. Use the mouse to select the column of [A ] data (G7:G27). X Values: Click in the Y Values box. Use the mouse to select the column of pH values. (C7:C27). Click Next. Chart Wizard Step 3: Enter appropriate graph titles. Click Finish. Your graph will now appear. Characteristics of a Buffer: A solutions composition would move along the x-axis by adding acid ( ) or base ( ). Question 1: How does the pH change when you add acid? (UP or DOWN) Question 2: If the [A ] is 1.0 mM and you add 0.7 mM of base, what is the change in pH? ____ Question 3: If the [A ] is 3.9 and you add 0.7 mM base, what is the change in pH? ____ Question 4: Which point is a better buffer? (1.0 mM A or 3.9 mM A ) Graph Cleanup: Take the time to fix the font and number format for the titles and scales. To do this, click on the item then use the font and formatting buttons the button bar. Excel tends to over emphasize labels making your actual data less prominent.

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11 Updated spreadsheet for disturbing the buffer system. Start by inserting eight rows below Buffer pK a. 1. Select the Excel row labels from Row 6 to 13. 2. Open the Insert menu and select Rows The column titles for the graph data should now be on Row 14. We will now add features to the spreadsheet so we can see the effect of adding base to the buffer. Labels: Add the labels on Rows 7-11 & 13. Note that we are labeling the graph section now to differentiate it from our new work. Concentrations: Enter initial buffer concentrations of 2.4 mM for [HA] and [A ]. This corresponds to the center of the buffer point. Enter 0.0 for the added base. We will change this number later. Calculations: Final [HA] is the initial minus [Added Base]: [HA] final = [HA] initial – [Added Base] Final [A ] is the initial plus [Added Base]. [A ] final = [A ] initial + [Added Base] Disturbed pH is calculated from the Henderson- Hasselbalch equation. Be sure to use [HA] final and [A ] final. Graphing: We will add a one-point series to the graph showing the Disturbed pH. 1. Right-Click on the graph, select Source Data. 2. Click on the Series tab, then click Add. 3. Set Name to Cell B11. 4. Set X Values to Cell F9. 5. Set Y Values to Cell E11. 6. Click Ok. Your graph should now have an extra magenta dot on it representing the disturbed pH. This dot will move along the graph as we modify the added base value. Investigating the Buffer: Simulate adding acid to the buffer by decreasing the value of [Added Base]. Adjust it until the Disturbed pH is equal to 7.40, the average physiologic pH. Question 5: How much acid relative to the overall buffer concentration do you have to add? Question 6: Is the blood buffer at an optimal point for resisting more base? Question 7: Is the blood buffer at an optimal point for resisting more acid? Justify your answers!

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12 Introduction: A buffer is a solution that resists changes in pH when acid or base is added. Buffer Capacity is an attempt to quantify this resistance. If you add a lot of acid or base, you can overwhelm the buffer! Definition and Meaning of Buffer Capacity What is Buffer Capacity? Older Definition: An older definition of buffer capacity was the amount of strong acid or base required to change 1 liter of the buffer solution by 1 pH unit. For biological buffers, this definition is not very useful. A change of 0.5 pH units can kill you. To study buffer capacity for smaller changes in pH, we interpret the buffer capacity definition as a ratio: Buffer Capacity = Amount of Acid 1 pH Unit Another Approach: The graph axes are pH and Acid Concentration. The slope of the graph would be: Slope = Change in pH Acid Amount This is the inverse of the definition of buffer capacity. The buffer capacity can be calculated as the inverse of the slope of our graph at a given point. (If youve had calculus, slope = derivative). Excel has an automatic function to calculate the slope for us. Calculating Buffer Capacity: 1.Label a new column next to your data BC 2.Enter the slope formula in Cell H16: =1/SLOPE(C15:C17,G15:G17) 3.Copy and paste this cell formula down the column through Cell H34. Note: We cannot calculate the slope for the first or last data point. Graphing Buffer Capacity 1. Right-Click on the graph, select Source Data, add another data series: Name = BC column label. X data = [A ] data. Y data = BC data. 2. Click on one of the BC dots to select the new data series. 3. Right Click on the same dot and then select Format Data Series. 4. Click on the Axis tab and select Secondary Axis. Meaning of Buffer Capacity: Question 8: Where is buffer capacity highest? Question 9: Is the blood pH at an optimal buffer capacity? Question 10: What is the value of the buffer capacity at the blood pH? The units would be (mM/pH unit).

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13 How much CO 2 do you lose per breath? Hyperventilation disturbs the blood pH by removing CO 2 from your blood faster than your metabolism produces it. –This is equivalent to adding hydroxide and disturbing the buffers pH. –On average, humans breathe at 24 breaths/minute (bpm). –For this exercise, assume that hyperventilation removes about 0.00060 mM of CO 2 per extra breath. –Based on this information, how long can you hyperventilate at 300% normal rate before reaching a dangerous blood pH? Solution: –Determine the amount of hydroxide you would have to add to the blood buffer to bring the pH from 7.40 to 7.80. –Use dimensional analysis to convert from mM base to extra breaths then minutes of hyperventilation:

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14 How much CO 2 do you lose per breath? We will now add features to the spreadsheet so we can calculate the time to death. Start by inserting 11 rows below the Buffer pK a row. Your Disturbing a Buffer section should now start on Row 18. Labels: Label this section Hyperventilation. Add the labels in Rows 11-16. Be sure to skip the entries in Rows 8-10, were going to cheat! Cheating: We need to determine [A ] at two different pH values. We could redo the second example buffer calculation… or we can cheat by using the formulas we created earlier. 1. Select Cells B25-G27. 2. Open the Edit menu and select Copy. 3. Click on Cell B8. 4. Open the Edit menu and select Paste. 5. Change the row labels to Normal and Death, and update the pH values. Calculations: 1. Enter a formula in Cell G11 to calculate the change in [A ]. 2. Enter constants for the unit conversions. 3. Calculate the extra breathing rate. 4. Apply dimensional analysis to calculate the time before death. Investigating Hyperventilation: Use (modify) your spreadsheet to answer the following questions. Question 11: How long would you last if you were really scared and hyperventilated at 100 bpm? Question 12: What if you watched a lame science fiction movie and only breathe at 32 bpm? Question 13: Is your answer to Question 12 realistic? Should a 33% higher breathing rate really be dangerous?

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15 I.Answer Questions 1-13 in the previous slides. II.Add a column to your spreadsheet to calculate the percent of conjugate base %A - = [A ] /([HA]+[A ])*100%. Modify your graphs so the x-axis is percent conjugate base. III.Make a copy of your spreadsheet. (Right-click on the Sheet1 tab at the bottom of the screen, select Move or Copy, check Create a Copy and click Ok.). Modify your spreadsheet for a primitive alien species that uses a phosphate buffer (pK a =7.2). Work through the spreadsheet process to answer the following questions: a. Find the amount of base needed to disturb the buffer to pH 7.40. b. Compare the buffer capacity this alien has to humans. c. How long can the aliens hyperventilate at 72 bpm? d. Would the aliens be more sensitive to acidosis? Explain how you used your spreadsheet to test this numerically. What boundary value will you use for pH? IV.We use a 0.025 M phosphate buffer to calibrate our electrodes (pH=7.00). If you had 50 mL of this buffer and didnt rinse the pH electrode, what effect would it have on the calibration buffer if: a. There was about 0.10 mL ( 2 drops) of neutral water on the electrode. b. There were 2 drops of 0.1M HCl on the electrode. End of Module Questions

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16 Pre-Post Test 1.If log(x)=3, what is x? 2.Use a calculator to determine the log of the numbers 0.05, 0.95, 1.0, and 5.3. Which of these values is special for logarithms? 3.Identify the point on the graph (A, B, or C) that best answers the following: a.Which point represents the lowest slope? b.Which point represents the solution composition that will best resist pH change (y-axis) when acid or base is added (move on the x-axis). 4.Use the Henderson-Hasselbalch equation (at right) to compute the value of pH in the equation for [HA]=1.2 mM and [A ]=24mM. 5.Which value ([HA] or [A ]) would you increase to lower the pH? 6.Compute the ratio [A ]/[HA] for a solution with pH=6.2. 7.Without doing any math, would the ratio increase or decrease for a pH of 7.2? A pH Added Base (mM) C B

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17 A spreadsheet is an easy way to perform calculations. The numbers in Cells B3 through B7 can just be typed in. As an alternative, Excel can do this for you. Type in the first three values then highlight them (B3 through B5) and place the cursor at the bottom right of the last highlighted cell until you see a small cross. Hold down the left mouse button, drag the pattern through as many cells as you want, and release the button to fill the cells. Excel recognizes the pattern from the first three cells and copies it. Getting Ready: Using a Spreadsheet – Data Input If you want to multiply each of these numbers by 2, you can create a recurrence formula to perform this task. In Cell C3, type the formula as shown. (All formulas begin with =.) You can copy the formula by clicking on Cell C3 and placing the cursor on the bottom right-hand corner of the cell until you see a small cross. Then drag the cursor down the column, and your results will be displayed. Slides 17-19 used with permission from module SSAC2006.Q199.CC1.2, Is It Hot in Here? -- Spreadsheeting Conversions in the English and Metric Systems by Cheryl Coolidge, http://serc.carleton.edu/sp/ssac/examples/14348.html.http://serc.carleton.edu/sp/ssac/examples/14348.html

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18 Getting Ready: Using a Spreadsheet – Calculation Input When the formula in Cell D3 is copied, the cell referenced in the numerator of the formula will adjust row by row, but the cell referenced in the denominator remains fixed. Suppose you always want to divide the numbers in Column C by the same number – lets use 10 for an example. You could create a formula for the first cell in Column C, =C3/10, and drag the formula down Column D as described in Slide 5. Suppose, though, that you might want to divide by a value in a particular cell. So that you dont have to change the formula for each value in Column C, you can reference the cell (here, C9) in the formula. In your formula, you refer to this cell as an absolute (or fixed) cell whose position doesnt change when you copy the formula. To indicate that this cell is absolute, precede both the column and the row number with a dollar sign. You can make a graph by highlighting a range of data (here, from B3 to C7) and then clicking on the chart wizard button: Select a graph type (in this case, an X-Y scatter plot connected by a smooth line) and follow the prompts. Voila! A graph!

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19 Getting Ready: Using a Spreadsheet – Number Formatting When working with percentages in Excel, it is best to treat them as decimals rather than values greater than 1 (e.g., 0.51 instead of 51%). Multiplying your decimals by 100 to obtain percents can, at times, needlessly complicate your equations and hinder Excels ability to understand what youre trying to calculate. To tell Excel to display the result as a percent, simply highlight the cells with your decimals, and follow the formatting directions previously discussed. However, instead of choosing Number from the Category list, choose Percentage, and select the number of additional decimal places you wish to use. Depending on the default settings of the version of Excel you are using, the values generated by your equations may display an unnecessary number of decimal places. To fix this, right-click on the cell or group of cells you wish to change and choose Format Cells from the pop- up menu. Select the Number tab, and choose Number from the Category list, if not already selected. In the Decimal places scroll box that appears on the right, type in the number of decimal places you would like to use. Click here to return to the buffer problem.

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20 Buffer Review: Derivation of the Henderson-Hasselbalch Equation Derivation (Click to start the steps): 1.Starting from the mass action equation, take the base-10 log of both sides. 2.Selectively separate the terms on the right-hand side (split off [H + ]). 3.Move log[H + ] to the left-hand side. 4.Move log(K a ) to the right-hand side. 5.The p in pH is an operator that means minus log or log 10 (). Replace log[H + ] with pH. 6.Replace log(K a ) with pK a. The Henderson-Hasselbalch equation is derived from the mass action equation for a weak acid. Chemical Equation: HA H + + A - Mass Action Equation: Click here to return to the buffer review.

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21 Buffer Review: Derivation of the Henderson-Hasselbalch Equation Click here to return to the buffer review.

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