Acids and bases · Lesson 2

A scale made of powers of ten

Pure water is never just H₂O. A tiny fraction of its molecules are always swapping protons, and that fixes the link between hydrogen ion and hydroxide in every solution. The pH scale turns those tiny numbers into ones you can use.

In 1909 the Danish chemist Søren Sørensen was working at the Carlsberg Laboratory in Copenhagen, the research institute founded by the brewery. He was studying how enzymes and proteins behave, and he kept running into the same problem: the concentration of hydrogen ions in his solutions could be 0.1 mol per litre in one flask and 0.000 000 1 in the next. Writing numbers like that all day is miserable, and comparing them is worse.

His solution was to use the power of ten instead of the number itself. A hydrogen ion concentration of 10⁻³ became simply “3”. He called it pH. (The p is usually said to stand for power, although historians still argue about it.)

More than a century later, pH is on shampoo bottles and swimming-pool test kits, and in every Year 12 chemistry exam. Behind it sits a simple equilibrium that is happening inside every glass of water you have ever drunk.

Water ionises itself

A tiny fraction of water molecules ionise into hydrogen ions and hydroxide ions:

H₂O(l) ⇌ H⁺(aq) + OH⁻(aq)

This self-ionisation happens to only about two molecules in every billion at 25 °C, and the ions recombine almost as soon as they meet. It is a dynamic equilibrium, and its equilibrium constant is the ionisation constant of water:

\[K_\mathrm{w} = [\mathrm{H^{+}}][\mathrm{OH^{-}}] = 1.0 \times 10^{-14} \quad \text{at 25 °C}\]

Liquid water is left out of the expression as usual. In pure water the ions are formed in pairs, so [H⁺] = [OH⁻] = 1.0 × 10⁻⁷ mol L⁻¹.

Loading the 3D model…

Water, up close

Watch water molecules transfer protons and the ions recombine. Add a little acid or base and see what happens to the other ion; then change the temperature.

K_w holds in every aqueous solution

Add acid to water and [H⁺] rises. The reverse reaction (H⁺ meeting OH⁻) speeds up, so hydroxide ions are used up until the product [H⁺][OH⁻] is back to 1.0 × 10⁻¹⁴. So at 25 °C, in any aqueous solution:

an acidic solution has [H⁺] greater than [OH⁻];
a neutral solution has them equal;
a basic solution has [OH⁻] greater than [H⁺];
you can always find one from the other: \([\mathrm{OH^{-}}] = \dfrac{K_\mathrm{w}}{[\mathrm{H^{+}}]}\).

For example, if [H⁺] = 2.0 × 10⁻³ mol L⁻¹, then [OH⁻] = 1.0 × 10⁻¹⁴ ÷ 2.0 × 10⁻³ = 5.0 × 10⁻¹² mol L⁻¹. Even a strongly acidic solution contains some hydroxide ions.

The pH scale

\[\mathrm{pH} = -\log_{10}[\mathrm{H^{+}}] \qquad [\mathrm{H^{+}}] = 10^{-\mathrm{pH}}\]

At 25 °C, pH 7 is neutral, below 7 is acidic and above 7 is basic. Because pH is a logarithm, each step of one unit is a factor of 10 in [H⁺]: a solution of pH 2 has 1000 times the hydrogen ion concentration of a solution of pH 5. pH can be negative or greater than 14 for very concentrated solutions.

Significant figures in pH

The number of decimal places in a pH equals the number of significant figures in the concentration. [H⁺] = 3.2 × 10⁻⁴ mol L⁻¹ (2 significant figures) gives pH = 3.49 (2 decimal places).

Calculating pH

Strong acids. A monoprotic strong acid is fully ionised, so [H⁺] equals the acid concentration. For 0.025 mol L⁻¹ HCl, \(\mathrm{pH} = -\log(0.025) = 1.60\).

Strong bases. Find [OH⁻] first (remember that Ba(OH)₂ gives two OH⁻ per formula unit), then use K_w. For 0.0050 mol L⁻¹ NaOH, \([\mathrm{H^{+}}] = \dfrac{1.0 \times 10^{-14}}{5.0 \times 10^{-3}} = 2.0 \times 10^{-12}\) mol L⁻¹, so pH = 11.70.

Dilution. The amount of acid is unchanged, so c₁V₁ = c₂V₂. Diluting a strong acid tenfold raises its pH by exactly one unit. But dilution can never push an acid past pH 7: once [H⁺] from the acid becomes tiny, the ions from water itself take over and the pH approaches 7.

Mixing an acid with a base. Work in moles. The H⁺ and OH⁻ neutralise each other one to one; whichever is in excess sets the pH, after dividing by the total volume.

A pH calculation checklist

Is the solution acidic or basic? For a base, find [OH⁻] first.
For mixtures, calculate moles of H⁺ and OH⁻, subtract, and divide the excess by the total volume in litres.
Convert to [H⁺] (using K_w if needed), take −log, and round to the right number of decimal places.
Sense check: an acid must have pH below 7 and a base above 7 (at 25 °C).

Temperature changes K_w

The self-ionisation of water is endothermic (ΔH ≈ +57 kJ mol⁻¹). From Lesson 4 of the equilibrium unit, K for an endothermic reaction increases with temperature, so K_w increases:

Temperature	K_w	pH of neutral water
0 °C	1.1 × 10⁻¹⁵	7.48
25 °C	1.0 × 10⁻¹⁴	7.00
37 °C	2.4 × 10⁻¹⁴	6.81
50 °C	5.5 × 10⁻¹⁴	6.63

So pure water at 50 °C has a pH of 6.63, yet it is not acidic. Neutral means [H⁺] = [OH⁻], and that is still true: both ions increased equally. pH 7 is the neutral point only at 25 °C.

Common traps

For a base, −log[OH⁻] is not the pH. Convert with K_w first.
Mixing two acid solutions does not average their pH values. Add the moles of H⁺ and divide by the total volume.
“pH less than 7 means acidic” is only true at 25 °C.

Now put it to workFive worked examples, twenty practice questions and ten harder problems on this lesson.

Go to the questions

Water ionises itself

Water, up close

Kw holds in every aqueous solution

The pH scale

Calculating pH

Temperature changes Kw

K_w holds in every aqueous solution

Temperature changes K_w