The formula
- pH from hydronium concentration
pH = -log[H₃O⁺]- Concentration from pH
[H₃O⁺] = 10^(-pH)- pOH and its link to pH at 25 °C
pOH = -log[OH⁻] and pH + pOH = 14.00- Water's ion product at 25 °C
K_w = [H₃O⁺][OH⁻] = 1.0 × 10⁻¹⁴
What the symbols mean
| Symbol | Meaning | Unit |
|---|---|---|
pH | Acidity measure; below 7.00 is acidic at 25 °C, above 7.00 is basic | dimensionless |
pOH | The same measure applied to hydroxide | dimensionless |
[H₃O⁺] | Molar concentration of hydronium ion, often written [H⁺] | M |
[OH⁻] | Molar concentration of hydroxide ion | M |
K_w | Ion product constant for water | 1.0 × 10⁻¹⁴ at 25 °C |
log | Base-10 logarithm, not the natural logarithm | dimensionless |
When it applies
- The pH + pOH = 14.00 relation and the neutral point at 7.00 both depend on K_w, which is 1.0 × 10⁻¹⁴ only at 25 °C. At other temperatures both shift.
- A strong acid is fully ionized, so its hydronium concentration equals its molarity and pH follows in one step. A weak acid needs its K_a and an equilibrium calculation first.
- Digits after the decimal point in a pH are the significant figures. pH 2.60 carries two significant figures, matching a concentration of 2.5 × 10⁻³ M.
- The simple treatment holds for solutions that are not extremely dilute; below about 10⁻⁶ M the self-ionization of water also has to be counted.
Worked example
Problem. A solution has [H₃O⁺] = 2.5 × 10⁻³ M at 25 °C. Find its pH, its pOH, and the hydroxide concentration.
- Apply the definition: pH = -log(2.5 × 10⁻³) = 2.602, which is pH 2.60 with two significant figures after the point.
- The value is below 7.00, so the solution is acidic, which fits a hydronium concentration well above 10⁻⁷ M.
- Get pOH from the 25 °C relation: pOH = 14.00 - 2.60 = 11.40.
- Find hydroxide from K_w: [OH⁻] = 1.0 × 10⁻¹⁴ / 2.5 × 10⁻³ = 4.0 × 10⁻¹² M.
- Check it: -log(4.0 × 10⁻¹²) = 11.40, matching the pOH from step 3.
Answer. pH 2.60, pOH 11.40, and [OH⁻] = 4.0 × 10⁻¹² M.
Common mistakes
- Using the natural logarithm. pH is defined with the base-10 log, and ln gives an answer 2.303 times too large.
- Dropping the minus sign, which turns an acidic pH of 2.60 into an impossible -2.60.
- Counting the digits before the decimal point as significant. In pH 2.60 only the 60 carries information about the concentration.
- Treating a weak acid's molarity as its hydronium concentration. A 0.10 M acetic acid solution is nowhere near pH 1.00, because only a small fraction of it ionizes.
Related formulas
- Titration:
n(titrant) = M(titrant) × V(titrant) - Henderson-Hasselbalch equation:
pH = pKa + log([A⁻] / [HA]) - Molarity formula:
M = mol solute / L solution