Half-life

Half-life formula

The half-life is T½ = ln 2 / λ = 0.693 / λ. A sample with T½ = 5 days has λ = 0.693 / 5 = 0.1386 per day, and half of it is left after 5 days.

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.

500 of 1000 left

Half-life5 dλ = 0.1386 per d

Mean life τ7.213 d1/λ = 1.44 T½

Left after 5 d50%N = 500 of 1,000

Activity now8.02 × 10⁻⁴ Bq2.17 × 10⁻¹⁴ Ci

Time to 50% left5 d1 half-lives

After 5 d, 50% is left: 500 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 = 5 / 5 = 1
N / N₀ = (½)ⁿ = 0.5, so N = 500 of 1,000
Time to 50% left: t = ln(N₀ / N) / λ = ln(2) / 0.1386 = 5 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 = 5 / 5 = 1
N / N₀ = (½)ⁿ = 0.5, so N = 500 of 1,000
Time to 50% left: t = ln(N₀ / N) / λ = ln(2) / 0.1386 = 5 d

Real nuclei

Half-lives and decay constants

Nucleus	Half-life	λ
Polonium-218	3.05 min	0.227 per min
Iodine-131	8.02 d	0.0864 per d
Cobalt-60	5.27 y	0.132 per y
Radium-226	1600 y	4.33 × 10⁻⁴ per y
Carbon-14	5730 y	1.21 × 10⁻⁴ per y
Uranium-238	4.47 × 10⁹ y	1.55 × 10⁻¹⁰ per y

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