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Half-life

Radioactive decay law

The radioactive decay law is N = N₀ exp(-λt), or N = N₀ (1/2)ⁿ after n half-lives. After three half-lives (15 days for T½ = 5 days) one eighth is left: 125 of 1,000 nuclei.

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.

125 of 1000 left
1T½2T½3T½4T½1½¼⅛N / N₀
Half-life5 dλ = 0.1386 per d
Mean life τ7.213 d1/λ = 1.44 T½
Left after 15 d12.5%N = 125 of 1,000
Activity now2.01 × 10⁻⁴ Bq5.42 × 10⁻¹⁵ Ci
Time to 25% left10 d2 half-lives

After 15 d, 12.5% is left: 125 of 1000 nuclei.

Show the working
  1. λ = 0.693 / T½ = 0.693 / 5 = 0.1386 per d
  2. Mean life τ = 1 / λ = 1.44 T½ = 7.213 d
  3. N = N₀ e-λt = N₀ (½)n with n = t / T½ = 15 / 5 = 3
  4. N / N₀ = (½)n = 0.125, so N = 125 of 1,000
  5. Time to 25% left: t = ln(N₀ / N) / λ = ln(4) / 0.1386 = 10 d
Working

Step by step

  1. λ = 0.693 / T½ = 0.693 / 5 = 0.1386 per d
  2. Mean life τ = 1 / λ = 1.44 T½ = 7.213 d
  3. N = N₀ e-λt = N₀ (½)n with n = t / T½ = 15 / 5 = 3
  4. N / N₀ = (½)n = 0.125, so N = 125 of 1,000
  5. Time to 25% left: t = ln(N₀ / N) / λ = ln(4) / 0.1386 = 10 d
Half-life by half-life

What is left

TimeFraction leftOf 1000
Start11,000
1 T½ = 5 days1/2500
2 T½ = 10 days1/4250
3 T½ = 15 days1/8125
4 T½ = 20 days1/1663
5 T½ = 25 days1/3231
Derivation

Why it works

  1. 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.
  2. Solving gives the radioactive decay law N = N₀ e-λt.
  3. After one half-life N = N₀/2, so e-λT½ = 1/2 and T½ = ln 2 / λ = 0.693 / λ.
  4. The average lifetime of a nucleus is τ = 1 / λ = T½ / 0.693 = 1.44 T½.
  5. After n half-lives N = N₀ (1/2)ⁿ: one half, one quarter, one eighth.
Questions

Radioactive decay law: quick answers

What is the radioactive decay law?

The number of nuclei left falls exponentially: N = N₀ exp(-λt), where λ is the decay constant. Its rate form is dN/dt = -λN.

How much is left after three half-lives?

One eighth: (1/2)³ = 1/8, or 12.5%. Of 1000 nuclei, 125 are left.

Can we say when one nucleus will decay?

No. Decay is random for a single nucleus; the law describes what happens on average to a large number of them.

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