What is a polyprotic acid?
A polyprotic acid can donate more than one proton. Common cases are
diprotic acids (\(\ce{H2A}\)) and triprotic acids (\(\ce{H3A}\)).
Ionization occurs in steps, each with its own constant
\(K_{\mathrm{a}1}\), \(K_{\mathrm{a}2}\), \(K_{\mathrm{a}3}\), with the usual trend
\(K_{\mathrm{a}1} \gt K_{\mathrm{a}2} \gt K_{\mathrm{a}3}\).
Stepwise ionizations (balanced equations)
\(\ce{H2A + H2O <=> H3O+ + HA-}\) with \(K_{\mathrm{a}1}\)
\(\ce{HA- + H2O <=> H3O+ + A^2-}\) with \(K_{\mathrm{a}2}\)
\(\ce{H3A + H2O <=> H3O+ + H2A-}\) with \(K_{\mathrm{a}1}\)
\(\ce{H2A- + H2O <=> H3O+ + HA^2-}\) with \(K_{\mathrm{a}2}\)
\(\ce{HA^2- + H2O <=> H3O+ + A^3-}\) with \(K_{\mathrm{a}3}\)
Key relations used by the calculator
- Step 1 (monoprotic form): for initial \(C_0\) and change \(x\),
\[
K_{\mathrm{a}1} \;=\; \frac{x^2}{C_0 - x}
\quad\Longrightarrow\quad
x \;=\; \frac{-K_{\mathrm{a}1} + \sqrt{K_{\mathrm{a}1}^{\,2} + 4K_{\mathrm{a}1}C_0}}{2}.
\]
- Step 2: with starting \([\ce{H3O+}]=H_1\) and \([\text{conj. acid}]=C_1\),
\[
K_{\mathrm{a}2} \;=\; \frac{(H_1+y)\,y}{C_1-y}
\quad\Longrightarrow\quad
y^{2}+y\!\left(H_1+K_{\mathrm{a}2}\right)-K_{\mathrm{a}2}C_1=0.
\]
- Step 3 (triprotic only): with starting \([\ce{H3O+}]=H_2\) and \([\text{conj. acid}]=C_2\),
\[
K_{\mathrm{a}3} \;=\; \frac{(H_2+z)\,z}{C_2-z}
\quad\Longrightarrow\quad
z^{2}+z\!\left(H_2+K_{\mathrm{a}3}\right)-K_{\mathrm{a}3}C_2=0.
\]
- pH / pOH (25 °C): \(K_{\mathrm{w}}=1.0\times10^{-14}\),
\(\mathrm{pH}=-\log_{10}[\ce{H3O+}]\),
\(\mathrm{pOH}=-\log_{10}[\ce{OH-}]\),
\([\ce{OH-}]=K_{\mathrm{w}}/[\ce{H3O+}]\), and \(\mathrm{pH}+\mathrm{pOH}=\mathrm{p}K_{\mathrm{w}}\approx14.00\).
Vertical ICE tables (concept)
The calculator uses vertical ICE tables for each step. Here is the generic layout for a diprotic acid:
|
\([\ce{H2A}]\) |
\([\ce{H3O+}]\) |
\([\ce{HA-}]\) |
| I | \(C_0\) | \(0\) | \(0\) |
|---|
| Δ | \(-x\) | \(+x\) | \(+x\) |
|---|
| E | \(C_0-x\) | \(x\) | \(x\) |
|---|
|
\([\ce{HA-}]\) |
\([\ce{H3O+}]\) |
\([\ce{A^2-}]\) |
| I | \(x\) | \(x\) | \(0\) |
|---|
| Δ | \(-y\) | \(+y\) | \(+y\) |
|---|
| E | \(x-y\) | \(x+y\) | \(y\) |
|---|
Common approximations (when \(K_{\mathrm{a}1}\gg K_{\mathrm{a}2}\gg K_{\mathrm{a}3}\))
- Most of \([\ce{H3O+}]\) is produced in the first ionization step:
\([\ce{H3O+}] \approx x\).
- For the second step, if \(y \ll x\), then \(y \approx K_{\mathrm{a}2}\) (since
\([\ce{H2A-}] \approx [\ce{H3O+}]\)).
- For a third step (triprotic), if \(z \ll y\),
\([\ce{A^3-}] \approx \dfrac{K_{\mathrm{a}3}[\ce{HA^2-}]}{[\ce{H3O+}]}\).
- Validity check (5% rule): If the percentage change is \(\le 5\%\), the small-\(x\) (or \(y,z\)) approximation is acceptable.
Special case — strong first step
Some acids have an essentially complete first ionization (e.g., \(\ce{H2SO4}\) at moderate concentrations).
In that case, take \(x\approx C_0\) for step 1 and solve only the second step using \(K_{\mathrm{a}2}\).
Worked examples (to compare with the calculator)
Phosphoric acid \(\ce{H3PO4}\), \(C_0=3.0\,\mathrm{M}\) — \(K_{\mathrm{a}1}=7.1\times10^{-3}\), \(K_{\mathrm{a}2}=6.3\times10^{-8}\), \(K_{\mathrm{a}3}=4.2\times10^{-13}\)
Using the exact quadratics step-by-step (as the calculator does) gives approximately:
- \([\ce{H3O+}] \approx 0.14\,\mathrm{M}\), \([\ce{H2PO4-}] \approx 0.14\,\mathrm{M}\)
- \([\ce{HPO4^2-}] \approx 6.3\times10^{-8}\,\mathrm{M}\)
- \([\ce{PO4^3-}] \approx 1.9\times10^{-19}\,\mathrm{M}\)
- \(\mathrm{pH} \approx 0.85\)
Sulfuric acid \(\ce{H2SO4}\), \(C_0=0.50\,\mathrm{M}\) — first step strong; \(K_{\mathrm{a}2}\approx 1.1\times10^{-2}\)
Treat step 1 as complete so \([\ce{H3O+}] \approx 0.50\,\mathrm{M}\) and \([\ce{HSO4-}] \approx 0.50\,\mathrm{M}\) before step 2. Solving the second
step with \(K_{\mathrm{a}2}\) yields roughly:
- \([\ce{H3O+}] \approx 0.51\,\mathrm{M}\), \([\ce{HSO4-}] \approx 0.49\,\mathrm{M}\)
- \([\ce{SO4^2-}] \approx 1.1\times10^{-2}\,\mathrm{M}\)
- \(\mathrm{pH} \approx 0.29\)
Notes & assumptions
- Activities are approximated by concentrations (ideal dilute solutions).
- \(K_{\mathrm{w}}=1.0\times10^{-14}\) and \(\mathrm{p}K_{\mathrm{w}}=14.00\) are assumed at \(25^{\circ}\mathrm{C}\).
- Very concentrated solutions or strong acid first steps may require ionic-strength corrections for high accuracy.
