Half-life

The half-life, t½, is the time taken for half the reactant to be used up.

For a first order reaction:

t½ = 0.693/k

Notice what is missing: [R]₀. The half-life of a first order reaction does not depend on how much you start with. Halving 100 g to 50 g takes exactly as long as halving 2 g to 1 g.

For a zero order reaction it is different: t½ = [R]₀/2k, so the half-life is proportional to the starting concentration. Start with more, and halving takes longer.

Because each half-life halves what is left, after n half-lives the fraction remaining is (½)ⁿ: a half after one, a quarter after two, an eighth after three.

Two handy results that follow from the first order equation:

• Reaching 99.9% completion takes about 10 half-lives.
• Going to 99% completion takes twice as long as going to 90%.

Radioactive decay is always first order. That is the basis of carbon dating. Living things keep taking in carbon, including a little radioactive carbon-14, so its share stays steady. When a plant or animal dies, the ¹⁴C is no longer topped up and decays with a half-life of 5730 years. Comparing the ¹⁴C left in old wood or bone with that in a living thing gives its age.