Half-Life — Tritium in Exit Signs (Worked Example)

Easy

Problem

A luminous exit sign contains 40 g of tritium (H-3) with half-life 12.3 years. How much tritium remains after 24.6 years?

Given

N₀ = 40 g
t½ = 12.3 years
t = 24.6 years

Find

The remaining mass N.

Principle

N = N₀(1/2)^(t/t½) — the exponent counts how many halvings occurred.

Formula

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

N = N₀(1/2)^(t/t½)

Solution

Half-lives elapsed: $n = 24.6/12.3 = 2$
Apply: $N = 40 \times (1/2)^2$
Calculate: $N = 40 \times 1/4 = 10\ \mathrm{g}$

10 g

Unit check

g × dimensionless = g — correct.

What this means physically

10 g — the sign glows dimmer as tritium decays (its beta particles excite the phosphor). After ~25 years it is at quarter brightness and needs replacement.

Common mistake

Using 24.6/12.3 = 2.5 by sloppy division. Count half-lives exactly: 24.6 is precisely 2 × 12.3.