The ratio of (¹⁴ C)/(¹² C) in a piece of wood is 1/8 part that of atmosphere. If half life of ¹⁴ C is 5730 years, the age of wood sample is ________ years.

Solution

Chemistry ma’am · 3 steps

One-eighth is (1/2)³, so the wood has lived through exactly three half-lives.

1. Why the ratio works. A living tree keeps swapping carbon with the air, so its ¹⁴C/¹²C ratio matches the atmosphere. Once it dies, ¹⁴C decays while ¹²C stays fixed. The falling ratio therefore tracks the ¹⁴C that is left.

   N/N₀ = (1/2)ⁿ = 1/8 ⇒ n = 3

2. Count half-lives. Each half-life halves the amount: 1 → 1/2 → 1/4 → 1/8. That is three halvings, so n = 3.
3. So. Age = n × t_(1/2) = 3 × 5730 = 17190 years.

The common slip: 45840. Treating 1/8 as 8 half-lives gives 8 × 5730 = 45840. The fraction 1/8 comes from only 3 halvings.

The idea to keep. When the fraction left is a neat power of 1/2, count the halvings and multiply by the half-life. Use t = 2.303/λ log N₀/N only for awkward fractions.

Answer from the NTA final answer key. Solution by PrepWiser.