Ion Product of Water

First determine the hydroxide concentration, then convert via Kw. Example: sodium hydroxide with c = 0.001 mol/L is completely dissociated, so [OH⁻] = 10⁻³ mol/L. With Kw = [H₃O⁺]·[OH⁻] = 10⁻¹⁴ mol²/L² it follows that [H₃O⁺] = 10⁻¹⁴/10⁻³ = 10⁻¹¹ mol/L and thus pH = 11. It is faster via the pOH: pOH = −log(10⁻³) = 3, and because pH + pOH = 14, pH = 14 − 3 = 11. Both routes are equivalent. For polyvalent bases such as Ba(OH)₂, remember to use twice the hydroxide amount: c(OH⁻) = 2·c(Ba(OH)₂).

The relation follows directly from the ion product. Taking the logarithm of Kw = [H₃O⁺]·[OH⁻] = 10⁻¹⁴ mol²/L² and multiplying by −1 turns the product into a sum: −log[H₃O⁺] − log[OH⁻] = 14, i.e. pH + pOH = pKw = 14. The 14 is nothing magical, just the numerical value of pKw at 25 °C. Practically this means: if you know one of the two quantities, you automatically know the other. If the hydronium concentration rises by a factor of 10, the hydroxide concentration falls by a factor of 10; the product stays constant. At other temperatures the sum shifts: at 37 °C, pH + pOH ≈ 13.6.

It is constant only at a fixed temperature. Kw is an equilibrium constant of the endothermic autoprotolysis 2 H₂O ⇌ H₃O⁺ + OH⁻ and therefore rises markedly with temperature: pKw is 14.0 at 25 °C, about 13.6 at body temperature (37 °C) and only around 12.3 at 100 °C. Within a solution at a given temperature, however, Kw holds everywhere, whether acidic, neutral or basic; that is exactly what makes it so useful. If you add acid, [H₃O⁺] rises and [OH⁻] falls correspondingly, so the product stays constant. In very concentrated solutions activities deviate from concentrations, and the simple numerical value then holds only approximately.

Because acidic is not defined by the number 7 but by comparing hydronium and hydroxide concentrations. Neutral means [H₃O⁺] = [OH⁻]. At 100 °C the autoprotolysis is stronger (pKw ≈ 12.3), so both concentrations rise to √Kw ≈ 7×10⁻⁷ mol/L; the pH of neutral water thereby drops to about 6.1. But since hydronium and hydroxide are still exactly equally abundant, the water is neutral, not acidic. The rule to remember: pH 7 is the neutral point only at 25 °C. This transfer question is a popular exam trap because it exposes blind memorization of the 7.

From the water itself: in the autoprotolysis one water molecule transfers a proton to another, 2 H₂O ⇌ H₃O⁺ + OH⁻. This equilibrium lies extremely far to the left; at 25 °C only 10⁻⁷ mol/L each of hydronium and hydroxide ions form, meaning only a single ion pair per roughly 555 million water molecules. Nevertheless this suffices for a measurable electrical conductivity, which Kohlrausch and Heydweiller demonstrated as early as 1894 on elaborately purified water. This residual conductivity is the experimental proof of autoprotolysis and the reason why completely non-conducting water does not exist. All pH calculations in aqueous solutions build on this equilibrium.