Equivalence Point vs End Point, and the Second Equivalence Point

The equivalence point is the moment in a titration when the moles of titrant added exactly equal the moles of analyte that can react with it. A polyprotic acid has one equivalence point per ionizable proton, so a diprotic acid has a first and a second equivalence point, and the second falls at exactly twice the titrant volume of the first. The end point is different: it is where the indicator changes color, which is chosen to land close to the equivalence point.

Equivalence point vs end point

The gap between the two is the titration error. A good indicator has a color-change range that falls on the steep part of the curve, so the error is a fraction of a drop.

Finding the equivalence point

For a monoprotic acid and a monobasic titrant, the equivalence point occurs when M_acid × V_acid = M_base × V_base. If 25.0 mL of 0.100 M acid is titrated with 0.100 M NaOH, equivalence is at 25.0 mL. On a curve it is the midpoint of the steepest vertical section. Its pH depends on the species left in the flask: 7 for a strong acid with a strong base, above 7 for a weak acid with a strong base (the conjugate base makes OH^-), and below 7 for a weak base with a strong acid.

The second equivalence point of a polyprotic acid

Each proton takes one equivalent of base. For a diprotic acid H₂A:

1. First equivalence point: H₂A has all become HA^-. The volume is V₁ = moles of H₂A / M_base.
2. Second equivalence point: HA^- has all become A^2-. The volume is V₂ = 2 × V₁.
3. First half-equivalence point (V₁/2): [H₂A] = [HA^-], so pH = pK_a1.
4. Second half-equivalence point (1.5 × V₁): [HA^-] = [A^2-], so pH = pK_a2.
5. At the first equivalence point, the species is amphoteric, and pH is about (pK_a1 + pK_a2) / 2. For an amino acid with no ionizable side chain, this is the isoelectric point.
6. At the second equivalence point, the species is the fully deprotonated base A^2-, so the pH is basic and set by its hydrolysis.

Worked example: glycine

Titrate 25.0 mL of 0.100 M fully protonated glycine (pK_a1 = 2.34, pK_a2 = 9.60) with 0.100 M NaOH.

• Moles of glycine: 0.100 M × 0.0250 L = 2.50 mmol. Each equivalence point needs 2.50 mmol of NaOH per proton.
• First equivalence point: 25.0 mL. pH is about (2.34 + 9.60) / 2 = 5.97.
• Second equivalence point: 50.0 mL.
• Half-equivalence points: pH = 2.34 at 12.5 mL and pH = 9.60 at 37.5 mL.

At 50.0 mL, the total volume is 75.0 mL, so [glycinate] = 2.50 mmol / 75.0 mL = 0.0333 M. Glycinate is a weak base with K_b = 10^-14 / 10^-9.60 = 10^-4.40 = 4.0 × 10^-5. Then [OH^-] = √(K_b × C) = √(4.0 × 10^-5 × 0.0333) = 1.15 × 10^-3 M, so pOH = 2.94 and pH = 11.06.

What "trial 1" means in a lab titration

In a lab, trial 1 is usually a rough run. You add titrant fairly quickly to find approximately where the color changes, so you know when to slow down. Later trials add titrant dropwise near that volume. The accepted result is the average of trials that agree closely (commonly within 0.1 mL), and the rough trial 1 is often left out of the average. A trial that overshoots the end point is discarded, not corrected.

Common traps

• Assuming the second equivalence point is at the same pH as the first. It is at twice the volume, and a very different pH.
• Using the wrong pK_a at a half-equivalence point. The first half-equivalence point gives pK_a1 and the second gives pK_a2.
• Forgetting dilution at the second equivalence point. Total volume has doubled, which matters when you compute a concentration.
• Treating the end point as exact. It is the indicator's color change, and only approximates the equivalence point.
• Expecting two clear jumps when the pK_a values are close. The two steps blur together unless the values differ by several units.

One-minute recap

Equivalence is stoichiometric and calculated, and the end point is observed. A diprotic acid has equivalence points at V₁ and 2V₁, with pK_a1 at V₁/2 and pK_a2 at 1.5V₁. The first equivalence pH is the average of the two pK_a values, and the second is set by the fully deprotonated base.