Topic 14 · page 2 of 18

pH and hydrogen ion concentration

pH is the negative logarithm of the H⁺ concentration. Each pH unit is a factor of ten, and you convert both ways with the lg button and its inverse.

13 min read14.314.414.5

What pH measures 14.3

The H⁺ concentration of everyday solutions covers an enormous range, from about 1 mol dm⁻³ in strong acid to 0.000 000 000 000 01 mol dm⁻³ in strong alkali. Writing those numbers out is awkward, and a graph of them is useless. The pH scale squeezes them onto a short ruler by using logarithms.

pH=−lg⁡ [HX+] \text{pH} = -\lg\,[\ce{H+}]

The concentration in square brackets is always in mol dm⁻³, but pH itself has no units.

A colour bar for the pH scale from 0 on the left, red, to 14 on the right, violet, with the hydrogen ion concentration at each even pH beneath it, from 1 mol per cubic decimetre at pH 0 to 10 to the minus 14 at pH 14. Everyday examples are placed above the bar.
The pH scale. Everyday pH values are typical and approximate.

Two facts follow from the logarithm.

From [H⁺] to pH 14.4

Press lg (sometimes labelled log), type the concentration, close the bracket, and press =. Change the sign.

Worked exampleConcentration to pH

Calculate the pH of a solution with [H⁺] = 0.0350 mol dm⁻³.

  1. pH = −lg [H⁺] = −lg (0.0350)
  2. lg (0.0350) = −1.456, so pH = 1.46

Worked exampleA small concentration

Calculate the pH of a solution with [H⁺] = 4.2 × 10⁻⁵ mol dm⁻³.

  1. pH = −lg (4.2 × 10⁻⁵)
  2. lg (4.2 × 10⁻⁵) = −4.377, so pH = 4.38

Check the answer against the scale. 0.0350 is between 0.1 (pH 1) and 0.01 (pH 2), so the pH must lie between 1 and 2. If it does not, you have a sign or an entry error.

From pH to [H⁺] 14.5

Undo the logarithm with the 10x button, which is usually the shifted lg key.

[HX+]=10−pH [\ce{H+}] = 10^{-\text{pH}}

Worked examplepH to concentration

Calculate [H⁺] for a solution of pH 3.40.

  1. [H⁺] = 10−3.4010^{-3.40}
  2. Press 10x, then −, then 3.40, then =: 3.98 × 10⁻⁴
  3. The pH has two decimal places, so give two significant figures: 4.0 × 10⁻⁴ mol dm⁻³

Worked exampleA very alkaline solution

Calculate [H⁺] for a solution of pH 12.60.

  1. [H⁺] = 10−12.6010^{-12.60} = 2.51 × 10⁻¹³
  2. Two decimal places in the pH, so two significant figures: 2.5 × 10⁻¹³ mol dm⁻³

Two sanity checks save marks. The answer must be smaller than 1 (unless the pH is negative), and a higher pH must give a smaller number.

Using the factor of ten 14.414.5

Because each unit is a factor of ten, some questions need no calculator.

Worked exampleComparing two solutions

Solution A has pH 2.0 and solution B has pH 5.0. How many times more H⁺ does A contain?

  1. The difference is 5.0 − 2.0 = 3.0 pH units.
  2. 10³ = 1000, and A has the lower pH.
  3. A contains 1000 times as much H⁺ as B.

Check yourself

  1. Define pH.
  2. Calculate the pH of a solution with [H⁺] = 0.0025 mol dm⁻³.
  3. Calculate the pH of a solution with [H⁺] = 6.3 × 10⁻⁴ mol dm⁻³.
  4. Calculate [H⁺] for a solution of pH 4.25.
  5. Calculate [H⁺] for a solution of pH 11.85.
  6. A sample of lemon juice has pH 2.5 and a sample of milk has pH 6.5. How many times more H⁺ is in the lemon juice?

Answers cover these until you have tried the questions

  1. pH is the negative logarithm to base 10 of the hydrogen ion concentration in mol dm⁻³: pH = −lg [H⁺].
  2. pH = −lg (0.0025) = 2.60
  3. pH = −lg (6.3 × 10⁻⁴) = 3.201, so 3.20
  4. [H⁺] = 10−4.2510^{-4.25} = 5.62 × 10⁻⁵, so 5.6 × 10⁻⁵ mol dm⁻³ (two decimal places in the pH, so two significant figures).
  5. [H⁺] = 10−11.8510^{-11.85} = 1.41 × 10⁻¹², so 1.4 × 10⁻¹² mol dm⁻³.
  6. The difference is 6.5 − 2.5 = 4.0 pH units, so the ratio is 10⁴. The lemon juice has 10 000 times as much H⁺ as the milk.