Half-life

Decay constant and mean life

The decay constant λ = 0.693 / T½ is the chance per unit time that a nucleus decays, and the mean life is τ = 1 / λ = 1.44 T½. For T½ = 5 days, λ = 0.1386 per day and τ = 7.21 days; after one mean life 1/e = 36.8% is left.

Radioactive decay

N = N₀ e^-λt, T½ = 0.693 / λ, mean life τ = 1 / λ and activity R = λN in becquerel. The same law fits a first order reaction, with k in place of λ. The dots are 1000 nuclei.

368 of 1000 left

Half-life5 dλ = 0.1386 per d

Mean life τ7.213 d1/λ = 1.44 T½

Left after 7.213 d36.8%N = 368 of 1,000

Activity now5.9 × 10⁻⁴ Bq1.6 × 10⁻¹⁴ Ci

Time to 36.79% left7.213 d1.44 half-lives

After 7.213 d, 36.8% is left: 368 of 1000 nuclei.

Show the working

λ = 0.693 / T½ = 0.693 / 5 = 0.1386 per d
Mean life τ = 1 / λ = 1.44 T½ = 7.213 d
N = N₀ e^-λt = N₀ (½)ⁿ with n = t / T½ = 7.213 / 5 = 1.443
N / N₀ = (½)ⁿ = 0.3679, so N = 368 of 1,000
Time to 36.79% left: t = ln(N₀ / N) / λ = ln(2.718) / 0.1386 = 7.213 d

Working

Step by step

λ = 0.693 / T½ = 0.693 / 5 = 0.1386 per d
Mean life τ = 1 / λ = 1.44 T½ = 7.213 d
N = N₀ e^-λt = N₀ (½)ⁿ with n = t / T½ = 7.213 / 5 = 1.443
N / N₀ = (½)ⁿ = 0.3679, so N = 368 of 1,000
Time to 36.79% left: t = ln(N₀ / N) / λ = ln(2.718) / 0.1386 = 7.213 d

Derivation

Why it works

Each nucleus has the same chance of decaying in each second, so the rate of decay is proportional to the number left: dN/dt = -λN.
Solving gives the radioactive decay law N = N₀ e^-λt.
After one half-life N = N₀/2, so e^-λT½ = 1/2 and T½ = ln 2 / λ = 0.693 / λ.
The average lifetime of a nucleus is τ = 1 / λ = T½ / 0.693 = 1.44 T½.
After n half-lives N = N₀ (1/2)ⁿ: one half, one quarter, one eighth.

Questions