Half-Life Formula

Half-life is the time required for half of the radioactive nuclei in a sample to decay.

The Formula

N = N_0\left(\frac12\right)^n

When to use: Radioactive samples do not lose the same amount each time; they lose the same fraction each time.

Quick Example

If a sample starts with 80 g and the half-life is 10 years, then 40 g remains after 10 years and 20 g remains after 20 years.

Notation

N_0

is the initial amount,

N

is the remaining amount,

n

is the number of half-lives elapsed, and

t_{1/2}

is the half-life period.

What This Formula Means

Half-life is the time required for half of the radioactive nuclei in a sample to decay.

Radioactive samples do not lose the same amount each time; they lose the same fraction each time.

Worked Examples

Example 1

medium

An ancient artifact has

\tfrac{1}{16}

of its original C-14. With C-14 half-life of

5730

years, how old is it?

Answer

22920\ \text{years}

First step

(1/2)^n = 1/16 \Rightarrow n = 4

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Example 2

medium

An isotope has

t_{1/2} = 12

years. After

36

years, what percent of original remains?

Example 3

medium

Strontium-90 has half-life

29

years. A contaminated site has

20\text{ Bq/g}

activity initially. What is the activity after

87

years?

Common Mistakes

Subtracting half of the original amount each time instead of half of what remains - Fix this by naming the substances or sample, checking "Am I using particle counts, nuclear charge, mass number, electron arrangement, or isotope notation to describe an atom or ion?", and attaching units, formulas, states, or evidence to the final statement. - Fix this by naming the substances or sample, checking "Am I using particle counts, nuclear charge, mass number, electron arrangement, or isotope notation to describe an atom or ion?", and attaching units, formulas, states, or evidence to the final statement.
Forgetting to count the number of half-lives before using the formula - Fix this by naming the substances or sample, checking "Am I using particle counts, nuclear charge, mass number, electron arrangement, or isotope notation to describe an atom or ion?", and attaching units, formulas, states, or evidence to the final statement. - Fix this by naming the substances or sample, checking "Am I using particle counts, nuclear charge, mass number, electron arrangement, or isotope notation to describe an atom or ion?", and attaching units, formulas, states, or evidence to the final statement.
Assuming a sample ever reaches exactly zero after a finite number of half-lives - Fix this by naming the substances or sample, checking "Am I using particle counts, nuclear charge, mass number, electron arrangement, or isotope notation to describe an atom or ion?", and attaching units, formulas, states, or evidence to the final statement. - Fix this by naming the substances or sample, checking "Am I using particle counts, nuclear charge, mass number, electron arrangement, or isotope notation to describe an atom or ion?", and attaching units, formulas, states, or evidence to the final statement.
Using half-life from a keyword alone - Signal words like atom, proton, neutron only point to a possible model; the substances and evidence must match too. - Fix this by naming the substances or sample, checking "Am I using particle counts, nuclear charge, mass number, electron arrangement, or isotope notation to describe an atom or ion?", and attaching units, formulas, states, or evidence to the final statement.

Why This Formula Matters

Half-Life lets students predict how much of a radioactive substance is left after any amount of time. It underpins carbon dating of fossils, dosing of medical tracers, and judging how long nuclear waste stays hazardous.

Frequently Asked Questions

What is the Half-Life formula?

Half-life is the time required for half of the radioactive nuclei in a sample to decay.

How do you use the Half-Life formula?

Radioactive samples do not lose the same amount each time; they lose the same fraction each time.

What do the symbols mean in the Half-Life formula?

$N_0$ is the initial amount, $N$ is the remaining amount, $n$ is the number of half-lives elapsed, and $t_{1/2}$ is the half-life period.

Why is the Half-Life formula important in Chemistry?

What do students get wrong about Half-Life?

Students often know a formula related to half-life but skip the recognition step: Am I using particle counts, nuclear charge, mass number, electron arrangement, or isotope notation to describe an atom or ion? That leads to a correct-looking substitution attached to the wrong chemical model.

What should I learn before the Half-Life formula?

Before studying the Half-Life formula, you should understand: radioactivity.

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Related Concepts

Radioactivity

Related Formulas

Radioactivity Formula