pH of a Weak Acid from Ka

For a weak acid HA the equilibrium expression Ka = x²/(C − x) is a quadratic, but when the acid barely dissociates the x in the denominator is negligible and it collapses to x = [H⁺] = √(Ka·C). Take 0.100 M acetic acid with Ka = 1.8 × 10⁻⁵: [H⁺] = √(1.8 × 10⁻⁶) = 1.34 × 10⁻³ M, so pH = 2.87. That is a thousand times less acidic than 0.100 M HCl at pH 1.00 — the whole difference between a weak acid and a strong one at the same concentration.

The approximation earns its keep but has a known failure mode. It assumes the dissociated fraction is under about 5%, which holds when C is at least a hundred times Ka. Push a fairly strong weak acid to low concentration — 0.0010 M chloroacetic acid, Ka = 1.4 × 10⁻³ — and the square-root shortcut overshoots badly; you must solve the quadratic. It also ignores water's own contribution to [H⁺], which starts to matter for very dilute or very weak acids where the predicted pH creeps above about 6.5.