Half-Life — Radioactive Decay Formula & Examples

Half-life is the time for a radioactive sample to fall to half its current amount. If one half-life passes, half remains. If two pass, a quarter remains. If three pass, an eighth remains.

For one isotope under the usual exponential radioactive decay model, the amount remaining is

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

where $N_0$ is the initial amount, $N(t)$ is the amount left after time $t$, and $T_{1/2}$ is the half-life. That is the main formula most students need.

What Half-Life Means In Physics

Half-life does not mean every atom survives exactly the same amount of time. It describes the average behavior of a large group of unstable nuclei.

That distinction matters. Radioactive decay is random for one nucleus, but a large sample shows a stable pattern. That is why the half-life formula works well for bulk decay calculations.

Half-Life Formula And Decay Constant

The most practical form is

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

Use this form when the half-life is already known.

You may also see radioactive decay written as

$$N(t) = N_0 e^{-\lambda t}$$

where $\lambda$ is the decay constant. For the same exponential decay model, the two forms are equivalent, and

$$T_{1/2} = \frac{\ln 2}{\lambda}$$

Use that relationship only in the usual exponential decay model for a single isotope with a constant decay probability per unit time.

Half-Life Example: How Much Remains After 15 Days?

Suppose a sample starts with $240$ mg of a radioactive isotope, and its half-life is $5$ days. How much remains after $15$ days?

Start by counting half-lives:

$$\frac{15}{5} = 3$$

So the sample has gone through three halvings:

$$240 \to 120 \to 60 \to 30$$

Using the formula gives the same result:

$$N(15) = 240 \left(\frac{1}{2}\right)^{15/5} = 240 \left(\frac{1}{2}\right)^3 = 30$$

After $15$ days, $30$ mg remains.

This is the fastest way to think about many half-life questions: count half-lives first, then apply repeated halving or the formula.

Common Half-Life Mistakes

Treating Decay As Linear

The sample does not lose the same amount each interval. It loses the same fraction each half-life. That is why the graph curves downward instead of forming a straight line.

Using Half-Life To Predict One Atom

Half-life cannot tell you when one particular nucleus will decay. It only describes the statistical behavior of many nuclei.

Forgetting The Model Condition

The standard half-life formula assumes exponential radioactive decay for one isotope. If a problem adds other production or loss processes, the simple formula may no longer apply by itself.

Mixing Time Units

If the half-life is in days, the time in the formula must also be in days. Unit mismatches are one of the most common calculation errors.

Where Half-Life Is Used

Half-life appears in nuclear physics, radiometric dating, nuclear medicine, environmental tracing, and radiation safety. In each case, the useful question is the same: how quickly does the amount of undecayed material fall over time?

Try Your Own Version

Try the same setup with a starting amount of $480$ mg and the same $5$-day half-life, or keep $240$ mg and change the time to $20$ days. If you want step-by-step feedback, try your own version in GPAI Solver.