Hydrolysis of ions — when salts change the pH
Salts whose ions are conjugates of weak acids/bases hydrolyze water and shift \( \mathrm{pH} \).
Strong–strong salts (e.g., NaCl) do not hydrolyze appreciably.
Anion of a weak acid (basic solution)
Let the salt supply the base \( \text{A}^- \) from the weak acid HA at initial concentration \(C_0\).
Hydrolysis: A− + H2O ⇌ HA + OH−.
The base constant is \(K_b = K_w/K_a(\text{HA})\).
\[
\begin{aligned}
K_b &= \frac{[\text{HA}][\text{OH}^-]}{[\text{A}^-]} \\
&= \frac{x\cdot x}{C_0 - x} \;=\; \frac{x^2}{C_0 - x}
\end{aligned}
\]
\[
\begin{aligned}
x^2 + K_b x - K_b C_0 &= 0, \qquad
x \;=\; \frac{-K_b + \sqrt{K_b^2 + 4K_b C_0}}{2}.
\end{aligned}
\]
Here \(x=[\text{OH}^-]\). Then \( \mathrm{pOH}=-\log_{10}x\) and \( \mathrm{pH}=14.00-\mathrm{pOH}\).
If \(x\le 0.05\,C_0\), the common approximation \(x\approx\sqrt{K_b C_0}\) is acceptable.
Cation of a weak base (acidic solution)
For \( \text{BH}^+ \) from weak base B: BH+ + H₂O ⇌ B + H₃O⁺.
The acid constant is \(K_a = K_w/K_b(\text{B})\) and the same algebra gives:
\[
\begin{aligned}
K_a &= \frac{[\text{B}][\text{H}_3\text{O}^+]}{[\text{BH}^+]}
= \frac{x^2}{C_0 - x}, \\
x &= \frac{-K_a + \sqrt{K_a^2 + 4K_a C_0}}{2}
\end{aligned}
\]
where \(x=[\text{H}_3\text{O}^+]\), so \( \mathrm{pH}=-\log_{10}x\) and \( \mathrm{pOH}=14.00-\mathrm{pH}\).
The \(x \approx \sqrt{K_a C_0}\) test applies analogously.
Notes
- Use \(K_w = 1.0\times 10^{-14}\) at 25 °C (temperature dependent).
- Very dilute or very concentrated solutions may require activity corrections (beyond this tool).