Half-life formula: formula, calculator and worked example

What the symbols mean

$N$: the amount remaining
$N_{0}$: the starting amount
$t$: the time elapsed
$T$: the half-life: the time for half of it to decay (same units as t)

When to use it

Use it for radioactive decay, carbon dating, drug elimination and anything described by a half-life. After one half-life half remains, after two a quarter, after three an eighth; the formula covers times in between, and solving for t dates a sample from how much is left.

Worked example

A sample of 80 mg has a half-life of 5 hours. How much is left after 15 hours? (N₀ = 80, t = 15, T = 5)

Start from the formula.
$N = N_{0} {(\frac{1}{2})}^{\frac{t}{T}}$
Substitute $N_{0}$ = $80$ , $t$ = $15$ , $T$ = $5$ and work it out.
$N = 80 {(\frac{1}{2})}^{\frac{15}{5}}$
$= 80 \cdot {0.5}^{3}$
$= 80 \cdot 0.125$
$= 10$

Answer N = 10 mg: three half-lives, so it halves three times

Common mistakes

Subtracting half the original each half-life (80, 40, 0). Each half-life halves what is *left*.
Using t and T in different units, such as days and years.
Assuming it ever reaches exactly 0. Mathematically the amount only approaches 0.

Half-life formula

Calculator

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What the symbols mean

When to use it

Worked example

Common mistakes