The relationship pH = -log[H₃O⁺] and the dilution of an acid

The formula and its inverse

The relationship that defines pH in terms of the concentration of H₃O⁺ ions is pH = -log[H₃O⁺] (where [H₃O⁺] is in mol/L). To find the concentration from the pH, we use the inverse relationship [H₃O⁺] = 10⁻pH.

Example: calculating a pH

A solution has an H₃O⁺ concentration of 1.0 × 10⁻² mol/L. Its pH is:

pH = -log(1,0 x 10^-2) = 2,0

Example: calculating a concentration

A solution has a pH of 4.0. Its H₃O⁺ concentration is:

[H3O+] = 10^-4,0 = 1,0 x 10^-4 mol/L

The effect of a 10-fold dilution

For a strong acid, diluting a solution by a factor of 10 (for example, by taking 10 mL of solution and making it up to 100 mL with water) reduces the H₃O⁺ concentration by a factor of 10, which INCREASES the pH by exactly 1 unit. Example: a solution with pH = 2.0 ([H₃O⁺] = 1.0 × 10⁻² mol/L), diluted 10-fold, has a concentration of 1.0 × 10⁻³ mol/L, and therefore a pH of 3.0.

Why this works this way

As the pH scale uses a decimal logarithm, dividing a concentration by 10 is equivalent to subtracting 1 from the logarithm, and therefore adding 1 to the pH (due to the negative sign). This is why the pH scale ‘compresses’ concentrations that vary over several orders of magnitude into pH differences that are easy to work with.

Common mistake

A frequent mistake is to think that diluting an acid ‘by a factor of 10’ divides the pH by 10 (for example, changing a pH of 2 to 0.2). This is incorrect: it is the CONCENTRATION that is divided by 10, whilst the pH simply increases by ONE unit.