Percent Ionization as a Function of Initial Concentration
This note explains the chemistry behind the calculator that predicts the percent ionization
of a weak monoprotic acid or a weak monofunctional base from its ionization constant
(\(K_\mathrm{a}\) or \(K_\mathrm{b}\)) and the initial concentration \(C_0\).
It uses a vertical ICE table, an exact quadratic solution for the equilibrium change \(x\),
and the usual small-\(x\) approximation as a quick check.
1) Definitions
- Percent ionization (acid):
\[
\%\text{ ionization} \;=\; 100 \cdot \frac{\left[\ce{H3O+}\right]_{\!\text{eq}}}{C_0}
\;=\; 100 \cdot \frac{x}{C_0}.
\]
For a base: \(100 \cdot \dfrac{\left[\ce{OH-}\right]_{\!\text{eq}}}{C_0}=100 \cdot \dfrac{x}{C_0}\).
- \(K_\mathrm{a}=10^{-\mathrm{p}K_\mathrm{a}}\), \(K_\mathrm{b}=10^{-\mathrm{p}K_\mathrm{b}}\).
- The calculator assumes 1:1:1 stoichiometry (HA ⇌ H\(_3\)O\(^+\)+A\(^-\) or B ⇌ BH\(^+\)+OH\(^-\)).
2) Vertical ICE table (weak acid)
Reaction (written with water explicitly): \(\;\ce{HA(aq) + H2O(l) <=> H3O^+(aq) + A^-(aq)}\).
|
\(\left[\ce{HA}\right]\) |
\(\left[\ce{H3O+}\right]\) |
\(\left[\ce{A-}\right]\) |
| I | \(C_0\) | \(0\) | \(0\) |
|---|
| Δ | \(-x\) | \(+x\) | \(+x\) |
|---|
| E | \(C_0-x\) | \(x\) | \(x\) |
|---|
Law of mass action:
\[
K_\mathrm{a} \;=\; \frac{\left[\ce{H3O+}\right]\!\left[\ce{A-}\right]}{\left[\ce{HA}\right]}
\;=\; \frac{x^2}{C_0 - x}.
\]
3) Exact solution and small-\(x\) approximation
Rearrange \(K_\mathrm{a}(C_0-x)=x^2\) to obtain a quadratic in \(x\):
\[
x^2 + K_\mathrm{a}\,x - K_\mathrm{a}\,C_0 = 0
\quad\Longrightarrow\quad
\boxed{\,x \;=\; \frac{-K_\mathrm{a} + \sqrt{K_\mathrm{a}^2 + 4\,K_\mathrm{a}\,C_0}}{2}\,}.
\]
Then \(\%\text{ ionization}=100\cdot \dfrac{x}{C_0}\).
Approximation (when \(x \ll C_0\)):
neglect \(x\) in the denominator to give \(K_\mathrm{a}\approx \dfrac{x^2}{C_0}\Rightarrow
x\approx \sqrt{K_\mathrm{a}\,C_0}\).
Consequently,
\[
\%\text{ ionization} \;\approx\; 100 \cdot \sqrt{\frac{K_\mathrm{a}}{C_0}},
\]
which shows that percent ionization increases as the solution is diluted (it scales with \(C_0^{-1/2}\)).
A common validity check is the \(5\%\) rule: if \(x/C_0 \le 0.05\), the approximation is acceptable.
4) Weak base case
Reaction: \(\;\ce{B(aq) + H2O(l) <=> BH^+(aq) + OH^-(aq)}\).
|
\(\left[\ce{B}\right]\) |
\(\left[\ce{BH+}\right]\) |
\(\left[\ce{OH-}\right]\) |
| I | \(C_0\) | \(0\) | \(0\) |
|---|
| Δ | \(-x\) | \(+x\) | \(+x\) |
|---|
| E | \(C_0-x\) | \(x\) | \(x\) |
|---|
\[
K_\mathrm{b} \;=\; \frac{\left[\ce{BH+}\right]\!\left[\ce{OH-}\right]}{\left[\ce{B}\right]}
\;=\; \frac{x^2}{C_0 - x}
\quad\Longrightarrow\quad
x \;=\; \frac{-K_\mathrm{b} + \sqrt{K_\mathrm{b}^2 + 4\,K_\mathrm{b}\,C_0}}{2}.
\]
Percent ionization (base) = \(100\cdot x/C_0\). The small-\(x\) trend and the \(5\%\) rule apply identically.
5) Using pH or pOH (alternative inputs)
- If pH is given: \(x=\left[\ce{H3O+}\right]_{\!\text{eq}}=10^{-\mathrm{pH}}\).
Then \(\%\) ionization \(=100\cdot x/C_0\) and
\[
K_\mathrm{a}=\frac{x^2}{C_0-x}.
\]
- If pOH is given: \(x=\left[\ce{OH-}\right]_{\!\text{eq}}=10^{-\mathrm{pOH}}\).
Then \(\%\) ionization \(=100\cdot x/C_0\) and
\[
K_\mathrm{b}=\frac{x^2}{C_0-x}.
\]
6) Common-ion and dilution effects (qualitative)
- Dilution (smaller \(C_0\)) increases percent ionization:
\(\%\approx 100\sqrt{K/C_0}\).
- A common ion (e.g., added \(\ce{A^-}\) or \(\ce{H3O+}\) for an acid) decreases ionization via Le Châtelier’s principle
(not included unless entered in initial concentrations for a more general ICE setup).
7) Assumptions and limits
- Solutions are sufficiently dilute that activities \(\approx\) concentrations; ionic-strength effects are ignored.
- Applies to weak monoprotic acids/bases with 1:1 stoichiometry; polyprotic systems require stagewise equilibria.
- Not for strong acids/bases (they are essentially fully ionized in typical dilute solutions).
