Half-Life — Iodine-131 Decay (Worked Example)

Easy

Problem

A hospital receives 100 g of iodine-131 for thyroid treatment. I-131 has a half-life of 8.0 days. How much remains after 24 days?

Given

N₀ = 100 g
t½ = 8.0 days
t = 24 days

Find

The remaining mass N.

Principle

Each half-life halves the sample: N = N₀(1/2)^(t/t½). Count the half-lives first.

Formula

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

N = N₀(1/2)^(t/t½)

Solution

Count half-lives: $n = t/t_{1/2} = 24/8 = 3$
Apply: $N = 100 \times (1/2)^3$
Calculate: $N = 100 \times 1/8 = 12.5\ \mathrm{g}$

12.5 g

Unit check

g × (dimensionless) = g — a mass, correct.

What this means physically

12.5 g — after 3 half-lives only 1/8 remains. Hospitals order I-131 fresh because it decays too fast to stockpile; after a month it is essentially gone.

Common mistake

Subtracting instead of halving (100 − 3×50). Radioactive decay is multiplicative — each half-life multiplies by ½, never subtracts.