Result

Radioactive decay

N (t) = N_{0} e^{- λ t}, t_{1/2} = \frac{ln 2}{λ}

Statement

The number of undecayed nuclei remaining after time t, and the half-life corresponding to decay constant λ.

Symbols and units

$N (t)$	number of undecayed nuclei at time t	—
$N_{0}$	initial number	—
$λ$	decay constant	s⁻¹
$t_{1/2}$	half-life	s

When it applies

A large sample of a single radioactive isotope, decaying independently and randomly.

Don't use when

Assumes a large sample of a single radioactive isotope decaying independently and randomly — a small sample shows statistical fluctuations around this smooth exponential prediction.

Derivation

Each nucleus decays independently with a fixed probability per unit time λ, so the decay rate is proportional to the number remaining:

\frac{d N}{d t} = - λ N

Integrating gives exponential decay; setting N = N₀/2 gives the half-life:

N (t) = N_{0} e^{- λ t}, t_{1/2} = \frac{ln 2}{λ}

Worked examples

Fraction remaining after n half-lives

\frac{N ( n t _{1/2} )}{N _{0}} = (\frac{1}{2})^{n}

After ten half-lives, under 0.1% of the original material remains — the fraction depends only on n, never on the starting amount.

Watch out for

Half-life does not depend on how much material is present or how long it has already been decaying. After n half-lives, exactly (1/2)ⁿ of the original material remains, regardless of the starting amount — after ten half-lives, under 0.1% remains.