Overlapping Polyprotic Equilibria

Introduction

The tidy stage-by-stage picture of polyprotic titrations depends on one condition: the stepwise acid constants must be well separated. When successive pKa values lie within one or two units of each other, the second proton starts to leave before the first has finished leaving. Buffer regions merge, intermediate equivalence points vanish and the intermediate species never dominates the solution. This page shows how to decide when stages overlap and what that means for calculations and titrations.

Core explanation

Distribution fractions overlap. For a diprotic acid H₂A, the fractions α₀ (H₂A), α₁ (HA⁻) and α₂ (A²⁻) depend on pH and the two constants. Each fraction is significant within roughly ±2 pH units of its governing pKa values. If pKa₁ and pKa₂ are 5 units apart, α₁ rises almost to 1 between them. If they are only 1–2 units apart, the curves for α₀ and α₂ cross over α₁ before it can grow large.

The maximum intermediate fraction. α₁ is largest at pH = ½(pKa₁ + pKa₂). At that pH,

α₁(max) = 1 ÷ (1 + 2 × 10^(−ΔpKa/2))

where ΔpKa = pKa₂ − pKa₁. With ΔpKa = 5, α₁(max) ≈ 0.994: HA⁻ is almost the only species, so a sharp first equivalence point exists. With ΔpKa = 1.4, α₁(max) ≈ 0.72: over a quarter of the acid is still H₂A or already A²⁻ at the best possible pH.

Consequences for titration curves. A distinct intermediate jump needs the intermediate species to dominate strongly. A common working guide is that successive pKa values should differ by about 4 units or more for a usable separate end point. When the gap is smaller, the titration curve shows one broad buffer region and a single clear jump after all protons have been removed.

Examples. Succinic acid (pKa₁ ≈ 4.2, pKa₂ ≈ 5.6) and citric acid (pKa ≈ 3.1, 4.8, 6.4) both titrate as if all their protons were removed together; citric acid shows one clear end point at three moles of hydroxide per mole of acid. Oxalic acid (pKa ≈ 1.3, 4.3) is more separated, but its first step is so strong that the first jump is poorly defined; it is normally titrated to the second end point. Phosphoric acid, with gaps of about 5 units, is the classic well-separated case.

Why pKa values can never be identical. Even two identical, non-interacting –COOH groups show a statistical gap: the first proton can leave from either of two sites, while the conjugate base can gain it on either of two sites. This gives pKa₂ − pKa₁ = log 4 ≈ 0.60. Electrostatic repulsion adds more, because removing a proton from a negative ion is harder. Groups far apart in a long chain therefore approach the 0.6 minimum.

Calculation consequences. For overlapping acids, the first-dissociation approximation and the amphiprotic average-of-pKa formula both lose accuracy. Reliable pH values then come from the full distribution-fraction equations combined with a charge balance, usually solved numerically.

Formulae

α₁ = K₁[H₃O⁺] ÷ ([H₃O⁺]² + K₁[H₃O⁺] + K₁K₂). At [H₃O⁺] = √(K₁K₂): α₁(max) = 1 ÷ (1 + 2√(K₂/K₁)).

Step-by-step reasoning

To judge whether stages overlap:

1. Find ΔpKa between the steps in question. 2. Calculate α₁(max) with the formula above. 3. If α₁(max) is above about 0.99, treat the stages as separate. 4. If it is well below this, expect merged buffer regions and a combined end point. 5. For accurate pH, abandon the simple approximations and use full speciation.

Visual explanation

Sketch the three distribution curves against pH. For well-separated constants, the α₁ curve forms a broad, flat-topped plateau reaching nearly 1. As the pKa values move together, the plateau shrinks into a low, rounded hump, with α₀ and α₂ curves overlapping it on both sides.

Real-world analogy

Picture two passengers leaving a bus at stops that are far apart: the bus clearly carries one passenger for a long stretch. If the stops are next to each other, there is barely a moment when exactly one passenger is aboard. Close pKa values are like adjacent bus stops.

Real-world example

Citric acid is used as a food acidulant and in citrate buffers. Because its three stages overlap, a citrate system buffers smoothly over a wide range, roughly pH 2.5 to 7, rather than in three separate narrow zones. This wide, continuous range is exactly why it is so popular in food and pharmaceutical formulations.

Why?

Why does a small ΔpKa lower the intermediate fraction? At the midpoint pH, the ratios [H₂A]/[HA⁻] and [A²⁻]/[HA⁻] both equal √(K₂/K₁). When K₂ approaches K₁ these ratios approach 1, so H₂A and A²⁻ are nearly as abundant as HA⁻.

Common misconception

"Each proton of a polyprotic acid gives its own end point." Only when successive constants differ enough. Overlapping acids give one combined end point, and the formula alone cannot tell you how many jumps appear.

Worked example

Question: Succinic acid has pKa₁ = 4.21 and pKa₂ = 5.64. Find the maximum fraction of hydrogen succinate and the pH at which it occurs.

Reasoning: ΔpKa = 1.43, so 10^(−0.715) = 0.193. α₁(max) = 1 ÷ (1 + 2 × 0.193) = 1 ÷ 1.386 = 0.72. It occurs at pH = ½(4.21 + 5.64) = 4.93.

Answer: About 72% at pH ≈ 4.9, so no clean first end point is possible.

Quick check

1. What is the smallest pKa gap expected for two identical, independent acidic groups on one molecule? Answer: About 0.60, which equals log 4, from the statistical factor.

Exam focus

Be ready to explain, using pKa values, why an acid such as citric acid gives one end point despite being triprotic. Quote the ΔpKa of roughly 4 needed for separate end points, and link small gaps to low intermediate fractions on a distribution diagram.

Advanced insight

Overlapping equilibria are described by macroscopic constants, which lump together several microscopic pathways. In molecules like amino acids, a proton may leave from different sites to give tautomeric forms; microscopic constants describe each site. Techniques such as NMR titration can resolve these, but a pH titration alone only yields the macroscopic values.

Summary

When successive pKa values are close, the pH ranges of the stepwise dissociations overlap. The intermediate species reaches only α₁(max) = 1 ÷ (1 + 2 × 10^(−ΔpKa/2)), buffer regions merge and intermediate end points disappear. A gap of about 4 pKa units is needed for separate end points, and a statistical minimum of about 0.6 always exists.

Practice questions

1. Calculate α₁(max) for a diprotic acid with ΔpKa = 3.0. Answer: 10^(−1.5) = 0.0316; α₁(max) = 1 ÷ (1 + 0.063) ≈ 0.94. 2. Citric acid is titrated with NaOH. How many moles of hydroxide per mole of acid are needed at the single clear end point? Answer: Three, because all three overlapping protons are removed before the pH jumps. 3. Why do the simple average-of-pKa and first-dissociation approximations fail for overlapping acids? Answer: They assume one species dominates at each stage, but with close constants several species coexist in significant amounts. 4. Give two reasons why pKa₂ exceeds pKa₁ for a dicarboxylic acid. Answer: The statistical factor (about 0.6 units) and the extra electrostatic attraction holding a proton on an already negative ion.