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1 This activity has been password protected to prevent modification. In order to request an unprotected version of this activity, contact

2 B B THE ACID-BASE DISTRIBUTION FUNCTIONS Objectives 1. To understand the origin of the acid-base fractional distribution functions. 2. To be able to use acid-base distribution functions to calculate the concentration of a species at a given ph. You can find alternative treatments of this material in Harris chapter 10, section 3 and chapter 11, section 5. You don t have to know or have read any of that to do this activity. Consider the general monoprotic weak acid HB which dissociates thus: HB H + + B and assume that neither HB nor B participate in any other reactions. Suppose further that C HB is the formal concentration of HB in the solution, if any was put in, and C B is the formal concentration of the salt of B (like NaB or KB) if any was put in. The symbol α B is defined as the fraction of C HB + C B that s in the B form. So if half of all the HB and B in the solution is in the B form, then αb is 0.5. If HB is really weak or the ph is pretty acidic for some reason it may be that only a tenth of all the HB and B is actually in the B form. In that case, αb would be 0.1. Another way to look at a B is that it s the ratio of [B ] to the total amount of HB and B in the system: [ B ] α B = C HB + C B α B is what your textbook refers to as α or the fraction of dissociation. If you know α B and you d like to know the molar concentration of B, that s pretty easy to get: [ B ]= ( C HB + C B )α B 10/12/ :42:00 PM

3 Page 2 distribution functions Problem. We d like an easy way to calculate α B that doesn t require us to get out all sorts of equilibrium equations and do gobs of algebra every time. 1. Write the K a expression for the equilibrium above 2. The formal concentrations, C HB and C B represent all the B present, whatever its acid-base form. Write a mass balance expression that relates these formal concentrations to the actual molar concentrations of molecules or ions in which B ends up. 3. Solve the mass balance expression for [HB] 4. Now substitute this expression for [HB] into the K a expression and solve for [B ]. 5. When you factor out C HB + C B on the right-hand side, what s left is α B B.

4 distribution functions Page 3 6. The pk a of acetic acid is Calculate α OAc, the fraction of acetic acid in the acetate ion form, at ph Suppose that I prepare a solution, maybe intended as a buffer, that s 0.2F in acetic acid and 0.1F in sodium acetate. Use the α OAc you obtained above to calculate the molar concentration of acetate ion, [OAc ], at ph 5.0.

5 Page 4 distribution functions The symbol α HB represents the fraction of the total of HB and B that s in the HB form, that is, undissociated acid. Think of it as: [ HB] α HB = C HB + C B For a monoprotic acid in a solution where HB and B don t react in any other way, the fraction in the HB form and the fraction in the B form must add up to 1.0: α B + α HB = So plug the general expression you got for α B into the relation above and solve for α HB. 9. Calculate α HOAc, the fraction of acetic acid in the undissociated HOAc form, at ph Use your α HB to calculate [HOAc] for a solution that s 0.2F in acetic acid (with no added acetate salt) at ph 5.0. These two little formulae for fraction of material in a particular ionic form are called acid-base distribution functions. They re useful for any monoprotic weak acid and any monoacidic weak base as long as you consider it as the base of a conjugate acid. You obtained them algebraically from the K a expression without making any assumptions regarding the relative size of any quantities, so they describe exactly the behavior of the acid or base. You can use them instead of the K a expression when solving equilibrium problems. They don t work if HB or B form a precipitate or a complex with something else in solution, so care is in order on that score. If the ionic strength is high they are technically correct only for activities and the activity coefficient must be used to obtain molar concentrations.

6 distribution functions Page 5 Now consider the diprotic acid H 2 B. Phthalic and oxalic acids are neutral examples of such acids, and we ve seen how the fully protonated form of alanine can be regarded as H 2 B +, which behaves the same way. In a solution of H 2 B there s the possibility of three different B- containing species: H 2 B, HB and B 2. There are also three ways we can add B to the solution: as the free acid, H 2 B, the half salt, NaHB (sodium bicarbonate and potassium hydrogen phthalate are common examples) and the full salt, Na 2 B. Suppose that we represent the formal concentration of the free acid, H 2 B, as C H2B, the formal concentration of the half-salt, NaHB as C HB, and the formal concentration of the full salt as C B. When preparing a solution of a diacid we may employ any one, two or all three of these compounds, depending upon what we re trying to accomplish. It s no sweat to write the two K a expressions and a mass balance expression for such a general solution (try it if you doubt your ability. It ll be good for you). When you solve them the way you did the HB case earlier you end up with the following distribution functions: α H 2 B = α HB = [ H + ] 2 [ H + ] 2 + K a1 H + [ ]+ K a1 [ ] [ H + ] 2 + K a1 H + K a1 H + [ ]+ K a1 α B 2 = [ H + ] 2 + K a1 H + K a1 [ ]+ K a1 These look pretty formidable, part of the reason I didn t ask you to derive them right here. The reason they re worthwhile is because they work for any diprotic acid. Not just oxalic acid, not just alanine, not just ethylenediammonium ion, but any diacid at all. They re all you have to know. You might consider it quite an undertaking to calculate α HB at, say, ph 7.0, and it will take some button-pushing on your calculator. But consider the alternative: solving three simultaneous equations in three unknowns every time you need to know [HB ] at ph 7.0. And that s what you d have had to do until just now.

7 Page 6 distribution functions So let s try it out. Somewhere around the end of May I ll dump 100 lb of baking soda (NaHCO 3 ) into my 100,000 gallon swimming pool at camp. The formal concentration of sodium bicarbonate comes out to be F (aren t you glad this isn t physics? You d have been asked to calculate that yourself). The ph is 7.6. H 2 CO 3 is a perfectly good diprotic acid as long as no CO 2 gets away, which it doesn t until you get well on the acid side of 7.0. For carbonic acid K a1 = 4.45 x 10 7 and K a2 = 4.69 x Calculate α CO3, the fraction of all the carbonate present that s in the CO 3 2 form. 11. Use the α CO3 you obtained above to calculate [CO 3 ] in my swimming pool (this is important because I also dump in a whole lot of Ca +2 and if [CO 3 ] gets too high, CaCO 3 will precipitate out, making the water murky and no fun to swim in). 12. Just for fun, calculate how much CO 2 is dissolved in this water (that is, [H 2 CO 3 ]). You already calculated the value of that nasty denominator at this ph and it hasn t changed any.

8 distribution functions Page 7 Now what about something like orthophosporic acid, H 3 PO 4? Looks like it s an example of a tri-protic acid, like H 3 B. Now there are four different B-containing forms with which to deal: H 3 B, H 2 B, HB 2 and B 3. Here are a couple of the distribution functions. [ ] 3 [ H + ] 3 + K a1 [ H + ] 2 + K a1 H + α H 3 B = H + [ ]+ K a1 K a 3 α H 2 B = K a1 [ H + ] 2 [ H + ] 3 + K a1 [ H + ] 2 + K a1 H + [ ]+ K a1 K a 3 Yeah, OK, I ran out of gas. You don t actually think these things are fun to type do you? But that doesn t need to be a problem. 13. See if you can guess what the denominator for the α HB distribution function is. 14. And now what s the term in the numerator? 15. So, supposing I make a phosphate buffer by slopping together 0.5F NaH 2 PO 4 and 0.5F K 2 HPO 4. The ph turns out to be 6.9. Phosphoric acid has K a1 = 7.11 x 10 3, K a2 = 6.32 x 10 8, and K a3 = 7.1 x Calculate the molar concentration of the hydrogen phosphate ion, [HPO 4 2 ] in this solution.

9 Page 8 distribution functions Exercise 1 Calculate the concentration of formate ion in a 0.01F solution of formic acid in which the ph has been adjusted to 4.0 with NaOH.

10 distribution functions Page 9 Exercise 2 Calculate the concentration of the zwitterionic form of the α-amino acid glutamine in a 0.02F solution at ph 3.0.

11 Page 10 distribution functions Exercise 3 A solution containing 0.05F ethylenediamine is adjusted to ph 10.0 with HCl. What is the actual molar concentration of the ethylenediamine form under these conditions?

12 distribution functions Page 11 Exercise 4 Pyridoxal-5-phosphate is a tetra-protic acid that s listed in the appendix of your textbook. Without concerning ourselves with its structure, let s simply represent it as H 4 Py. Derive the distribution function for α Py, the fraction of the material present that s actually in the fullydeprotonated form, Py 4. You can do this from first principles if you d like with the four K a expressions and a massive mass balance, but it will do (and you ll be more likely to get the correct result) if you just figure it out by analogy using the other examples shown in this activity.