Half-Life — Uranium-238 Over Geologic Time (Worked Example)

Given

• N₀ = 1.0 kg

• t½ = 4.5 × 10⁹ years

• t = 9.0 × 10⁹ years

Find

The remaining mass N.

Principle

Even over billions of years the same halving rule applies: N = N₀(1/2)^(t/t½).

Formula

Solution

1. Half-lives: $n = 9.0 \times 10^9/(4.5 \times 10^9) = 2$

2. Apply: $N = 1.0 \times (1/2)^2$

3. Calculate: $N = 0.25\ \mathrm{kg}$

Unit check

Kilograms × (1/2)^n = kilograms. Halving is dimensionless, so mass units survive.

What this means physically

0.25 kg — the other 0.75 kg became lead-206 plus helium. Geologists read this parent/daughter ratio in rocks as a clock dating Earth's oldest minerals.