Hydrogen ion concentrations range from about 1 mol/L down to \mathrm{10^{-14}} mol/L, so it is easier to use the power of ten. When \mathrm{[H^+]} is \mathrm{10^{-n}} mol/L, the pH is n.
In other words, pH is minus the logarithm to base 10 of \mathrm{[H^+]}. A solution in which \mathrm{[H^+]} is \mathrm{10^{-3}} mol/L has a pH of 3.
Each step of 1 in pH is a factor of 10 in \mathrm{[H^+]}, so at pH 3 the hydrogen ion concentration is one tenth of that at pH 2. At 25 °C, pH 7 is neutral, lower values are acidic and higher values are basic.
For a base, first find \mathrm{[OH^-]}. Sodium hydroxide ionizes completely, so in 0.01 mol/L sodium hydroxide solution \mathrm{[OH^-]} is 1.0 × \mathrm{10^{-2}} mol/L.
The ionic product of water then gives \mathrm{[H^+]} = 1.0 × \mathrm{10^{-14}} / (1.0 × \mathrm{10^{-2}}) = 1.0 × \mathrm{10^{-12}} mol/L, so the pH is 12.
In the same way, 0.1 mol/L sodium hydroxide solution has a pH of 13, and the more concentrated a base, the higher its pH.
Lemon juice has a pH of about 2 and vinegar about 2.5, because both contain acids. Coffee is weakly acidic at about pH 5.
Clean rainwater has a pH of about 5.6, not 7, because carbon dioxide from the air dissolves in it and forms a weak acid.
Blood is kept at about pH 7.4 and seawater is about 8.1, while limewater, a solution of calcium hydroxide, is strongly basic at about 12.4.
Two acids of the same concentration can have different pH values. For an acid that gives one hydrogen ion per molecule, \mathrm{[H^+]} is the concentration of the acid multiplied by its degree of ionization.
In 0.1 mol/L hydrochloric acid, \mathrm{[H^+]} = 0.1 × 1 = 0.1 mol/L, so the pH is 1.0. In 0.1 mol/L ethanoic acid, \mathrm{[H^+]} = 0.1 × 0.013 = 1.3 × \mathrm{10^{-3}} mol/L, so the pH is about 2.9.
The weak acid has a hydrogen ion concentration about 77 times lower, even though it contains just as much acid, so its pH is about 2 units higher.
Diluting a strong acid ten times lowers \mathrm{[H^+]} ten times, so the pH rises by 1: hydrochloric acid of 0.1, 0.01 and 0.001 mol/L has a pH of 1, 2 and 3.
This cannot go on for ever. Water itself provides hydrogen ions at about \mathrm{10^{-7}} mol/L, so in very dilute acid these ions become important.
An acid of \mathrm{10^{-8}} mol/L does not have a pH of 8, which would be basic, but a pH of about 6.98. However much an acid is diluted, its pH approaches 7 but never goes past it.