Titration Curves (Polyprotic)
Polyprotic titration context
Polyprotic acids have multiple dissociation steps and therefore multiple buffer regions and equivalence points. This page provides practical pH approximations along titration progression using Henderson-Hasselbalch sections and equivalence-point approximations.
\[ \mathrm{pH} = \mathrm{p}K_{a,i} + \log\frac{n_{\mathrm{Base},i}}{n_{\mathrm{Acid},i}} \]
Typical use cases: carbonate, phosphate, and multi-step organic acid titrations.
- Estimate useful titration windows
- Identify dominant buffer zones
- Interpret multiple curve inflection points
Formulas (MathJax)
\[ n_{A,0}=c_A\,V_A \]
\[ n_B = c_B\,V_B \]
\[ V_{\mathrm{eq},j} = \frac{j\,n_{A,0}}{c_B} \quad (j=1,2,3) \]
\[ \mathrm{pH}_{\mathrm{buffer},i} \approx \mathrm{p}K_{a,i}+\log\frac{n_B-(i-1)n_{A,0}}{i n_{A,0}-n_B} \]
\[ \mathrm{pH}_{\mathrm{Eq},1} \approx \frac{\mathrm{p}K_{a1}+\mathrm{p}K_{a2}}{2},\quad \mathrm{pH}_{\mathrm{Eq},2} \approx \frac{\mathrm{p}K_{a2}+\mathrm{p}K_{a3}}{2} \]
Symbol legend
- \(c_A\): initial acid concentration [mol/L]
- \(V_A\): initial acid volume [L or mL consistent]
- \(c_B\): titrant base concentration [mol/L]
- \(V_B\): added base volume [L or mL consistent]
- \(n_{A,0}\): initial acid amount
- \(n_B\): added base amount
- \(\mathrm{p}K_{a1},\mathrm{p}K_{a2},\mathrm{p}K_{a3}\): stepwise acid constants
- \(V_{\mathrm{eq},j}\): j-th equivalence point volume
Detailed examples
Example 1 (diprotic H₂A):
\(c_A=0.1\,\mathrm{mol/L}\), \(V_A=25\,\mathrm{mL}\), \(c_B=0.1\,\mathrm{mol/L}\).
Equivalence points at \(25\,\mathrm{mL}\) and \(50\,\mathrm{mL}\).
At \(V_B=12.5\,\mathrm{mL}\): half-equivalence 1, so \(\mathrm{pH}\approx\mathrm{p}K_{a1}\).
\(c_A=0.1\,\mathrm{mol/L}\), \(V_A=25\,\mathrm{mL}\), \(c_B=0.1\,\mathrm{mol/L}\).
Equivalence points at \(25\,\mathrm{mL}\) and \(50\,\mathrm{mL}\).
At \(V_B=12.5\,\mathrm{mL}\): half-equivalence 1, so \(\mathrm{pH}\approx\mathrm{p}K_{a1}\).
Example 2 (second buffer range):
At \(V_B=37.5\,\mathrm{mL}\) (between first and second equivalence), pair \(\mathrm{HA^-}/\mathrm{A^{2-}}\) dominates, controlled by \(\mathrm{p}K_{a2}\).
At \(V_B=37.5\,\mathrm{mL}\) (between first and second equivalence), pair \(\mathrm{HA^-}/\mathrm{A^{2-}}\) dominates, controlled by \(\mathrm{p}K_{a2}\).
Example 3 (triprotic H₃A):
Three equivalence points occur at \(V_{eq,1},V_{eq,2},V_{eq,3}\).
Between them, three characteristic buffer zones appear with the corresponding \(\mathrm{p}K_a\) values.
Three equivalence points occur at \(V_{eq,1},V_{eq,2},V_{eq,3}\).
Between them, three characteristic buffer zones appear with the corresponding \(\mathrm{p}K_a\) values.
Example 4 (method note):
If \(\mathrm{p}K_a\) values are close, buffer zones overlap strongly and practical inflection points can become less distinct.
If \(\mathrm{p}K_a\) values are close, buffer zones overlap strongly and practical inflection points can become less distinct.
Deeper insight
Approximation limits
The formulas here provide robust practical approximations. For highly dilute systems or high ionic strength, activity corrections and full equilibrium modeling are recommended.
Labor interpretation
Polyprotic curves support indicator selection, endpoint-window definition, and potentiometric method planning. Where dissociation steps overlap, informed curve interpretation is critical.
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