Overview
A polyprotic acid donates more than one proton in steps:
\[
\mathrm{H_nA}\ \rightleftharpoons\ \mathrm{H}_{n-1}\mathrm{A^-} + \mathrm{H^+}\quad(K_{a1}),\qquad
\mathrm{H}_{n-1}\mathrm{A^-}\ \rightleftharpoons\ \mathrm{H}_{n-2}\mathrm{A^{2-}} + \mathrm{H^+}\quad(K_{a2}),\ \ldots
\]
Typically \(K_{a1} \gg K_{a2} \gg K_{a3}\). In a titration with strong base, the net reaction is
HnA(aq) + n OH⁻(aq) → An−(aq) + n H₂O(l)
Key volumes and moles
n0 = Ca·Va (moles of acid sites in the first step), with volumes in L.- Equivalence volumes (in mL) for base concentration \(C_b\):
\(\displaystyle V_{e,1} = \frac{C_a V_a}{C_b}\cdot 1000\),
\(V_{e,2}=2V_{e,1}\) (diprotic/triprotic),
\(V_{e,3}=3V_{e,1}\) (triprotic).
- Halfway points: \(H_1=\tfrac{1}{2}V_{e,1}\), \(H_2=\tfrac{V_{e,1}+V_{e,2}}{2}\), \(H_3=\tfrac{V_{e,2}+V_{e,3}}{2}\).
pH in each region (diprotic H₂A and triprotic H₃A)
Initial solution (before any base): for a weak first step,
\[
[\mathrm{H^+}] \approx \frac{-K_{a1} + \sqrt{K_{a1}^2 + 4K_{a1}C_a}}{2},\quad \mathrm{pH}=-\log_{10}[\mathrm{H^+}].
\]
Buffer regions (Henderson–Hasselbalch; use moles form)
- Before 1st equivalence \((0 < n_{\mathrm{OH^-}} < n_0)\): mixture of acid/conjugate base of step 1.
\[
\mathrm{pH} \approx \mathrm{p}K_{a1} + \log_{10}\!\left(\frac{n_{\text{base form}}}{n_{\text{acid form}}}\right)
= \mathrm{p}K_{a1} + \log_{10}\!\left(\frac{n_{\mathrm{H}_{n-1}\mathrm{A^-}}}{n_{\mathrm{H_nA}}}\right).
\]
- Between 1st and 2nd equivalence (diprotic/triprotic):
\[
\mathrm{pH} \approx \mathrm{p}K_{a2} + \log_{10}\!\left(\frac{n_{\mathrm{A^{2-}\ or\ H\!A^{2-}}}}{n_{\mathrm{H A^-}\ or\ H_2\!A}}\right).
\]
- Between 2nd and 3rd equivalence (triprotic):
\[
\mathrm{pH} \approx \mathrm{p}K_{a3} + \log_{10}\!\left(\frac{n_{\mathrm{A^{3-}}}}{n_{\mathrm{H A^{2-}}}}\right).
\]
- Halfway points: at \(H_1, H_2, H_3\) the ratio is 1, so
\(\mathrm{pH}\approx \mathrm{p}K_{a1},\ \mathrm{p}K_{a2},\ \mathrm{p}K_{a3}\) respectively.
Equivalence points
- Diprotic (H₂A):
- 1st equivalence (H A⁻ amphiprotic):
\(\displaystyle \mathrm{pH} \approx \frac{\mathrm{p}K_{a1} + \mathrm{p}K_{a2}}{2}\) (valid when \(\mathrm{p}K_{a1}-\mathrm{p}K_{a2}\gtrsim 2\)).
- 2nd equivalence (A²⁻ basic hydrolysis): with \(C_{\text{salt}}=\dfrac{n_0}{V_{\text{tot}}}\) and \(K_b=\dfrac{K_w}{K_{a2}}\),
\[
[\mathrm{OH^-}] = \frac{-K_b + \sqrt{K_b^2 + 4K_b C_{\text{salt}}}}{2},\quad \mathrm{pH} = 14 - \mathrm{pOH}.
\]
- Triprotic (H₃A):
- 1st equivalence (H₂A⁻ amphiprotic): \(\displaystyle \mathrm{pH} \approx \frac{\mathrm{p}K_{a1} + \mathrm{p}K_{a2}}{2}\).
- 2nd equivalence (HA²⁻ amphiprotic): \(\displaystyle \mathrm{pH} \approx \frac{\mathrm{p}K_{a2} + \mathrm{p}K_{a3}}{2}\).
- 3rd equivalence (A³⁻ basic hydrolysis): \(K_b=K_w/K_{a3}\) and the same quadratic as above for \([\mathrm{OH^-}]\).
After the last equivalence
Excess strong base dominates:
\[
[\mathrm{OH^-}] = \frac{n_{\mathrm{OH^-}} - n_{\text{needed}}}{V_{\text{tot}}},\qquad
\mathrm{pH}=14 - \big(-\log_{10}[\mathrm{OH^-}]\big).
\]
When are these approximations valid?
- Well-separated steps: \(\Delta\mathrm{p}K_a \gtrsim 2\) (ideally ≥ 3) between successive steps.
- Buffer formula (H–H) used when \(0.1 \lesssim \dfrac{n_{\text{base}}}{n_{\text{acid}}} \lesssim 10\).
- Amphiprotic estimates assume \(K_{a,\,i} \gg K_{a,\,i+1}\).
- This tool uses \(K_w=1.0\times10^{-14}\) (25 °C) and accounts for dilution, i.e., \(V_{\text{tot}}=V_a+V_b\).
What the calculator computes
- Determines the titration stage from moles \(n_{\mathrm{OH^-}}\) vs. \(n_0, 2n_0, 3n_0\).
- Applies the appropriate analytic formula:
initial weak-acid expression, Henderson–Hasselbalch (moles form),
amphiprotic average, or exact quadratic for salt hydrolysis.
- Corrects concentrations using the current
Vtot, and reports
key points (0 mL, halfways, equivalences, ±1 mL), your custom points, and plots the curve.
Quick reference
RegionModelpH expression
-
Start
Weak acid (1st step)
\( [\mathrm{H^+}] = \dfrac{-K_{a1}+\sqrt{K_{a1}^2+4K_{a1}C_a}}{2} \)
-
Buffer k
H–H (moles)
\( \mathrm{pH}=\mathrm{p}K_{a,k}+\log_{10}\!\dfrac{n_{\text{base}}}{n_{\text{acid}}} \)
-
Amphiprotic
Average
\( \mathrm{pH}\approx \dfrac{\mathrm{p}K_{a,i}+\mathrm{p}K_{a,i+1}}{2} \)
-
Salt at eq
Hydrolysis
\( [\mathrm{OH^-}]=\dfrac{-K_b+\sqrt{K_b^2+4K_b\,C_{\text{salt}}}}{2},\quad K_b=\dfrac{K_w}{K_a} \)
-
Excess base
Strong base
\( \mathrm{pH}=14+\log_{10}[\mathrm{OH^-}] \)
Tip: you can enter either \(K_a\) or \(\mathrm{p}K_a\). The graph highlights half-equivalence (\(\mathrm{pH}\approx\mathrm{p}K_a\)),
equivalences, and custom volumes. For closely spaced \(\mathrm{p}K_a\) values, real curves may show merged steps and require full
speciation; this tool follows standard analytical approximations used in general chemistry.
