Numford
Half-life

Carbon dating: age from carbon-14

Carbon-14 has a half-life of 5730 years. A sample with 25% of its carbon-14 left has gone through two half-lives, so it is t = ln(1/0.25) / λ = 11,460 years old.

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.

250 of 1000 left
1T½2T½3T½4T½1½¼⅛N / N₀
Half-life5730 yλ = 1.21 × 10⁻⁴ per y
Mean life τ8267 y1/λ = 1.44 T½
Left after 11460 y25%N = 250 of 1,000
Activity now9.58 × 10⁻¹⁰ Bq2.59 × 10⁻²⁰ Ci
Time to 25% left11460 y2 half-lives

After 11460 y, 25% is left: 250 of 1000 nuclei.

Show the working
  1. λ = 0.693 / T½ = 0.693 / 5730 = 1.21 × 10⁻⁴ per y
  2. Mean life τ = 1 / λ = 1.44 T½ = 8267 y
  3. N = N₀ e-λt = N₀ (½)n with n = t / T½ = 11460 / 5730 = 2
  4. N / N₀ = (½)n = 0.25, so N = 250 of 1,000
  5. Time to 25% left: t = ln(N₀ / N) / λ = ln(4) / 1.21 × 10⁻⁴ = 11460 y
Working

Step by step

  1. λ = 0.693 / T½ = 0.693 / 5730 = 1.21 × 10⁻⁴ per y
  2. Mean life τ = 1 / λ = 1.44 T½ = 8267 y
  3. N = N₀ e-λt = N₀ (½)n with n = t / T½ = 11460 / 5730 = 2
  4. N / N₀ = (½)n = 0.25, so N = 250 of 1,000
  5. Time to 25% left: t = ln(N₀ / N) / λ = ln(4) / 1.21 × 10⁻⁴ = 11460 y
Age from the fraction left

Carbon-14 left and age

C-14 leftHalf-livesAge
100%00 years
50%15,730 years
25%211,460 years
12.5%317,190 years
10%3.3219,035 years
1%6.6438,069 years
Derivation

Why it works

  1. Living things keep taking in carbon, so the share of carbon-14 in them matches the air.
  2. After death no new carbon-14 comes in and it decays with a half-life of 5730 years.
  3. Measure the fraction f still left: f = N / N₀ = (1/2)t / T½.
  4. So the age is t = T½ × log₂(1/f) = ln(1/f) / λ, with λ = 0.693 / 5730 per year.
Questions

Carbon dating: quick answers

How does carbon dating work?

Living things keep the same share of carbon-14 as the air. After death it decays with a half-life of 5730 years, so the fraction left gives the time since death.

How old is a sample with 25% of its carbon-14?

25% is (1/2)², two half-lives, so it is 2 × 5730 = 11,460 years old.

How far back can carbon dating go?

About 50,000 years. After nine or ten half-lives too little carbon-14 is left to measure well.

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