Ch. 9: Polyprotic Acid-Base Equilibria

Outline:

• 9-1 Diprotic Acids and Bases

• 9-2 Diprotic buffers

• 9-3 Polyprotic Acids and Bases

• 9-4 Which is the Principal Species?

• 9-5 Fraction Compositions

• 9-6 Isoelectric and Isoionic pH’s

Updated Oct. 24, 2011: new slides 1-24

Amino AcidsAmino acids are the building blocks of proteins, and always have an acidic carboxylic acid group, a basic amino group, and a neutral substituent, R.

At low pH: both the -COO and -NH2 groups are protonatedAt high pH: both the -COO and -NH2 groups are not protonated

There are separate pKa values associated with proton dissociation from all three of the functionalities (see table next page).

AA’s can also exist in zwitterionic form. Zwitterions are species carrying both positive and negative charges, which are stabilized, in the cases of AA’s by interactions of the -COO- and -NH3+ groups with water. In fact, AA’s in the solid state almost always have their most stable state as a zwitterion; however, in the gas phase, zwitterions rarely exist.

pKa’s of Amino Acids

LeucineOur discussion will focus upon Leucine (HL) (i.e., R is the isobutyl group, (CH3)2CHCH2—)

The associated equilibrium constants are:

(note: sometimes the subscripts ‘a’ are omitted on Ka1 and Ka2)

Recall: relations between these coefficients involve Kw:

Ka1iKb2 = Kw Ka2 iKb1 = Kw

We will examine three cases, where we calculate the pH’s and compositions of separate solutions of (i) 0.0500 M H2L+, (ii) 0.0500 M HL and (iii) 0.0500 M L-.

We note that they do not depend on the charge type of the acids and bases, meaning that we use the same procedure to find the pH of the diprotic H2A, where A is anything, or H2L+, where HL is leucine.

Leucine: Acidic form, H2L+

Leucine hydrochloride contains the protonated species, H2L+, which can dissociate twice. Since K1 = 4.70 × 10−3, H2L+ is a weak acid, and HL is an even weaker acid, with K2 = 1.80 × 10−10. We can make the approximation that a solution of H2L+ behaves as a monoprotic acid, with Ka = K1 (since H2L+ dissociates only partly, and HL much less).

Ka = K1 = 4.70 ×10−3

x2

F − x= Ka ⇒ x = 1.32 ×10−2M = [HL] = [H+]

[H+] = x = 1.32 ×10−2 ⇒ pH = 1.88[H2L

+ ] = F − x = 3.68 ×10−2M

We have assumed that [L] is very small; nonetheless, we can still calculate it.

K2 =[H+][L− ][HL]

⇒ [L− ] = K2[HL][H+]

= 1.80 ×10−10M

The approximation [H+] ≈ [HL] reduces the equation above to [L−] = K2 (good approximation!)

For most diprotic acids, K1 is sufficiently larger than K2 for this approximation to be valid. Even if K2 were just 10 times less than K1, [H+] calculated by ignoring the second ionization would be in error by only 4%. The error in pH would be only 0.01 pH unit. In summary, a solution of a diprotic acid behaves like a solution of a monoprotic acid, with Ka = K1.

Leucine: Basic form, L-

The species L−, found in a salt such as sodium leucinate, can be prepared by treating leucine (HL) with an equimolar quantity of NaOH. There are two basic species with Kb1 = 5.55 × 10−5 and Kb2 = 2.13 × 10−12. We can make the approximation that a solution of L- behaves as a monoprotic base, with Kb = Kb1 (L- does not really hydrolyze with H2O to give HL).

[H2L+] = Kb2 = 2.13 × 10−12 M, so the approximation that [H2L+] is insignificant relative to [HL] is justified. In summary, if there is any reasonable separation between Ka1 and Ka2 (and, therefore, between Kb1 and Kb2), the fully basic form of a diprotic acid can be treated as monobasic, with Kb = Kb1.

CO2 in the air and oceanDissolved carbon dioxide is one of the most important diprotic acids in Earth’s ecosystem. Increasing atmospheric CO2 increases the concentration of CO2 dissolved in the ocean, which consumes carbonate and lowers the pH:

The pH of the ocean has already decreased from its preindustrial value of 8.16 to 8.04 today, and is predicted to be 7.7 by 2100. Low [CO32-] promotes dissolution of solid CaCO3:

If CO32- in the ocean decreases enough, organisms such as plankton and coral with CaCO3 shells or skeletons will not survive. Calcium carbonate has two crystalline forms called calcite and aragonite (more soluble).When pteropods collected from the subarctic Pacific Ocean are kept in water that is less than saturated with aragonite, their shells begin to dissolve within 48 h. Animals such as the pteropod lie at the base of the food chain.

Leucine: Intermediate form, HLThis is more complicated to treat, since HL may act as either an acid or a base.

The HL is said to be amphiprotic (i.e., it can accept and/or donate a proton). However, we expect that this solution will be acidic, since Ka > Kb. But, we cannot simply ignore the hydrolysis reaction, even if there is a fairly large difference between Ka and Kb.

Since the first reaction produces H+, it reacts with the OH- in the second reaction to form H2O, thereby driving this reaction to the right. Hence, we need to treat this problem with the systematic treatment of equilibrium.

We have Step 1 listed above. Step 2 is the charge balance equation:

There are a number of substitutions we can make to ultimately calculate the species concentrations and the pH (written example). Summary for all three forms:

Simplified calculation: Intermediate formStarting from the equation we worked out using the S.T.E., we can make a few more simplifications (i.e., (i) K2F >> Kw and (ii) K1 >> F):

[H+ ] = K1K2F + K1Kw

K1 + F≈

K1K2FK1 + F

≈K1K2FF

= K1K2

Approx. (i) Approx. (ii)

Then, reframe the equation in terms of pH:

log[H + ] ≈ 12(logK1 + logK2 )

− log[H + ] ≈ −12(logK1 + logK2 )

pH ≈12(pK1 + pK2 )

Recall:

This very powerful and useful equation can be used in many situations, and says that the pH of the intermediate form of a diprotic acid is close to midway between pK1 and pK2, regardless of the formal concentration.

Simplified calculation: Intermediate form, 2Example:

Summary of diprotic acid calculationsSolution of H2A Solution of HA- Solution of HA

Treat H2A as a monoprotic acid with Ka = K1 to find [H+], [HA−], and [H2A].

Use the K2 equilibrium to solve for [A2−].

Treat HA- as a monoprotic base with Kb = Kb1 = Kw/Ka2 to find[H+], [HA−], and [A2-].

Use the K1 equilibrium to solve for [H2A].

Use the approximation [HA−] ≈ F and find the pH

With [H+] from step 1 and[HA−] ≈ F, solve for [H2A] and [A2−], using the K1 and K2 equilibria.

We must be careful, since we have not considered other equilibria (e.g., Na+ or K+ in solutions of HA− or A2− form weak ion pairs that we have neglected).

Diprotic BuffersA buffer made from a diprotic (or polyprotic) acid is treated in the same way as a buffer made from a monoprotic acid. For the acid H2A, we can write two Henderson-Hasselbalch equations, both of which are always true. If we happen to know [H2A] and [HA−], then we will use the pK1 equation. If we know [HA−] and [A2−], we will use the pK2 equation.

pH = pK1 + log [HA− ][H2A]

; pH = pK2 + log [A2− ][HA]

Polyprotic acids and basesThe treatment of diprotic acids and bases can be extended to polyprotic systems.

1. H3A is treated as a monoprotic weak acid, with Ka = K1.

2. H2A- is treated as an intermediate form of a diprotic acid.

[H+ ] = K1K2F + K1Kw

K1 + F

3. HA2- treated as the intermediate form of a diprotic acid. However, HA2− is “surrounded” by H2A− and A3−, so the equilibrium constants to use are K2 and K3, instead of K1 and K2.

[H+ ] = K2K3F + K2Kw

K2 + F

4. A3− is treated as monobasic, with Kb = Kb1 = Kw/Ka3.

Polyprotic acids and bases, 2Example (written solution).

We have reduced acid-base problems to just three types. When you encounter an acid or base, decide whether you are dealing with an acidic, basic, or intermediate form. Then do the appropriate arithmetic to answer the question at hand.

Principal SpeciesWe sometimes must identify which species of acid, base, or intermediate predominates under given conditions. e.g., “What is the principal form of benzoic acid at pH = 8?”

At pH = 4.20, there is a 1:1 mixture of benzoic acid (HA) and benzoate ion (A−).At pH = pKa + 1 = 5.20, the quotient [A−]/[HA] is 10:1.At pH = pKa + 2 = 6.20), the quotient [A−]/[HA] is 100:1.As pH increases, the quotient [A−]/[HA] increases further.

Hence, for a monoprotic system:

The basic species, A−, is the predominant formwhen pH > pKa.

The acidic species, HA, is the predominant form when pH < pKa.

Thus, the predominant form of benzoic acid at pH 8 is the benzoate anion, C6H5CO2-.

Principal Species, 2For polyprotic systems, our reasoning is similar, but there are several values of pKa. Consider oxalic acid, H2Ox, with pK1 = 1.25 and pK2 = 4.27:At pH = pK1, [H2Ox] = [HOx−].At pH = pK2, [HOx−] = [Ox2−].The chart shows the major species in each pH region.

Hence, for a polyprotic system:

The basic species, A2-, is the predominant formwhen pH > pK2.

The intermediate species, HA-, is the predominant form when pK1 < pH < pK2

The acidic species, H2A, is the predominant form when pH < pK1.

Principal Species, 3What is the principal form of arginine at pH 10.0? Approximately what fraction is in this form?What is the second most abundant form at this pH?

At pH = pK2 = 8.99,[H2Arg+] = [HArg].At pH = pK3 = 12.1[HArg] = [Arg−].

At pH = 10.0, the major species is HArg.

Since pH 10.0 is about one pH unit higher than pK2, we can say [HArg]/[H2Arg+] ≈ 10:1. i.e., ca. 90% of arginine is in the form HArg. The second most important species is H2Arg+, which makes up ca. 10% of the arginine.

Fractional CompositionsWe now derive equations that give the fraction of each species of acid or base at a given pH. These equations are useful for acid-base and EDTA titrations, and electrochemical equilibria.Monoprotic systemsGoal: Find an expression for the fraction of an acid in each form (HA and A−) as a function of pH. We can do this by combining the equilibrium constant with the mass balance:

Rearranging the MB gives [A−] = F − [HA], which is plugged into the Ka expression to give

This can be rearranged to give the concentration of HA:

The fraction of molecules in the form HA is called αHA:

Fractional Compositions, 2If we divide the previous equation by F, we get:

αHA =[HA]F

=[H+]

[H+]+ Ka

Fraction in the form HA

αA− =[A− ]F

=Ka

[H+]+ Ka

Fraction in the form A-

Here we see plots of αHA and αA- as a function of pH for a system with a pKa of 5.00.

At low pH, almost all of the acid is in the form HA.

At high pH, almost everything is in the form A−.

Note, that: αHA + αA- = 1.

The fraction here denoted as αA- is the same as the fraction of dissociation discussed earlier.

Fractional Compositions, 3Diprotic SystemsThe derivations for the fractional composition equations for diprotic systems is similar to that for the monoprotic system (don’t worry about this - let’s just look at the equations):

αH2A=[H2A]F

=[H+]2

[H+]2 + [H+]K1 + K1K2

αHA− =[HA− ]F

=K1[H

+][H+]2 + [H+]K1 + K1K2

αA2− =[A2− ]F

=K1K2

[H+]2 + [H+]K1 + K1K2

Fractional composition diagram for fumaric acid (trans-butenedioic acid). At low pH, H2A is dominant. At intermediate pH, HA− is dominant; and, at high pH, A2− dominates. Because pK1 and pK2 are not separated very much, the fraction of HA− never gets very close to unity.

Note: These equations can be applied equally to bases, where (K1 = Kw/Kb2 andK2 = Kw/Kb1)

Isoelectronic and Isoionic pHIn biochemistry, polyprotic molecules are often discussed in terms of the isoelectric or isoionic pH. Let’s consider a simple diprotic system: alanine.

The isoionic point (or isoionic pH) is the pH obtained when the pure, neutral polyprotic acid HA (the neutral zwitterion) is dissolved in water (i.e. the pH of the pure acid)Ions: H2A+, A−, H+ and OH−

Principal Species: HA, [H2A+] ≠ [A−]

The isoelectric point (or isoelectric pH) is the pH at which the average charge of the polyprotic acid is 0.Principal Species: uncharged form HA, [H2A+] = [A−]Equilibria: Always some H2A+ + A− ⇌ HA.

Isoelectronic and Isoionic pH, 2When alanine is dissolved, the pH is the isoionic pH. Since HA is the intermediate form of the diprotic acid, H2A+, then the [H+] is given by

[H+ ] = K1K2F + K1Kw

K1 + FIsoionic point:

For 0.10 M alanine, the isoionic pH is

From [H+], K1 and K2, [H2A+] = 1.68 × 10−5 M and [A−] = 1.76 × 10−5 M for pure alanine in water (the isoionic solution). There is a slight excess of A− because HA is a slightly stronger acid than it is a base. It dissociates to make A− a little more than it reacts with water to make H2A+.

The isoelectric point is the pH at which [H2A+] = [A−], and, therefore, the average charge of alanine is 0. To go from the isoionic solution (pure HA in water) to the isoelectric solution, we could add just enough strong acid to decrease [A−] and increase [H2A+] until they are equal. Adding acid necessarily lowers the pH. For alanine, the isoelectric pH must be lower than the isoionic pH.

Isoelectronic and Isoionic pH, 3Calculate the isoelectric pH by first writing expressions for [H2A+] and [A−]:

Setting [H2A+] = [A−], we find

which yields

pH = 12

(pK1 + pK2 )Isoelectronic point:

For a diprotic amino acid, the isoelectric pH is halfway between the two pKa values. The isoelectric pH of alanine is 0.5(2.34 + 9.87) = 6.10

The isoelectric and isoionic points for a polyprotic acid are almost the same. At the isoelectric pH, the average charge of the molecule is 0; thus [H2A+] = [A−] and pH = 0.5(pK1 + pK2). At the isoionic point, the pH is given by a more complex equation (see previous slide), and [H2A+] is not exactly equal to [A−].

Isoelectronic FocusingIsoelectronic focusing is a technique used for separation of proteins. At the isoelectronic point, the average charge of all proteins is zero. The application of a strong electric field will not result in migration of these proteins if they are at their isoelectronic pH’s.

However, if a medium is used in which there is a gradient of pH values, proteins migrate (positive to negative poles and negative to positive poles) until they reach regions of their isoelectronic pH’s. (i.e., Each protein is focused in one region at its isoelectric pH).

Lab-on-a-chip isoelectric focusing: 6-mm-long × 100 μm-wide × 25-μm-deep capillary etched into silica glass.

If a molecule diffuses out of its isoelectric region, it becomes charged and immediately migrates back to its isoelectric zone.

(i) Fluorescent pH markers.(ii) and (iii) Separation of fluorescence-labeled proteins: (OVA) ovalbumin; (GFP) green fluorescent protein; (BSA) bovine serum albumin; (Tfer) transferrin; (CA) carbonic anhydrase; (PhB) phosphorylase B; and (Hb) hemoglobin.

[G. J. Sommer, A. K. Singh, and A. V. Hatch, “On-Chip Isoelectric Focusing Using Photopolymerized Immobilized pH Gradients,” Analyt. Chem. 2008, 80, 3327.]