Half-Life — Dating with Carbon-14 (Worked Example)

Easy

Problem

A wooden artifact contains carbon-14 at 1/8 the concentration found in living wood. Carbon-14 has a half-life of 5730 years. How old is the artifact?

Given

N/N₀ = 1/8
t₁/₂ = 5730 years

Find

Age t of the sample.

Principle

Each half-life halves the remaining nuclei: after n half-lives the fraction left is (1/2)ⁿ.

Formula

N = N_0\left(\tfrac{1}{2}\right)^{t/t_{1/2}}

N = N₀(1/2)^(t/t₁/₂)

Solution

Set the fraction: $(1/2)^n = 1/8$
Recognize $1/8 = (1/2)^3$ , so $n = 3$ half-lives
Multiply: $t = 3 \times 5730 = 17{,}190\ \mathrm{years}$

≈ 17,200 years old

Unit check

Half-lives are counts; years × count = years. Consistent.

What this means physically

About 17,200 years — the artifact dates to the last ice age. Carbon dating works precisely because the half-life is a fixed nuclear clock, unaffected by temperature, pressure or chemistry.

Common mistake

Assuming the decay is linear ('1/8 gone means 1/8 of a half-life'). Radioactive decay is exponential — each half-life removes half of what remains.