Radioactive decay formula

Radioactive decay is a random process at the level of single atoms, but for a large number of atoms it follows a simple exponential law. For activity \(A\), number of atoms \(N\), or mass \(m\), the decay law has the same form:

\[ A(t) = A_0 e^{-\lambda t} \] \[ N(t) = N_0 e^{-\lambda t} \] \[ m(t) = m_0 e^{-\lambda t} \]

Here:

\(A_0, N_0, m_0\) – initial activity, atoms, or mass at time \(t = 0\)
\(A(t), N(t), m(t)\) – value after time \(t\)
\(\lambda\) – decay constant (units of 1/time)
\(t\) – elapsed time

Half-life and decay constant

The half-life \(T_{1/2}\) is the time it takes for the activity (or number of atoms, or mass) to drop to half its initial value. The relationship between half-life and decay constant is:

\[ T_{1/2} = \frac{\ln 2}{\lambda} \quad\Longleftrightarrow\quad \lambda = \frac{\ln 2}{T_{1/2}} \]

The fraction remaining after time \(t\) is:

\[ \frac{A(t)}{A_0} = e^{-\lambda t} = \left(\frac{1}{2}\right)^{t/T_{1/2}} \]

Solving for different unknowns

Depending on what you know and what you want to find, you can rearrange the decay law:

Given \(A_0, \lambda, t\) → find \(A(t)\): \[ A(t) = A_0 e^{-\lambda t} \]
Given \(A(t), \lambda, t\) → find \(A_0\): \[ A_0 = A(t) e^{\lambda t} \]
Given \(A_0, A(t), t\) → find \(\lambda\): \[ \lambda = -\frac{1}{t}\ln\left(\frac{A(t)}{A_0}\right) \]
Given \(A_0, A(t), \lambda\) → find \(t\): \[ t = -\frac{1}{\lambda}\ln\left(\frac{A(t)}{A_0}\right) \]

The same equations apply if you replace activity \(A\) with number of atoms \(N\) or mass \(m\).

Worked example

Example: A sample of carbon‑14 has an initial activity of 1000 Bq. The half-life of ¹⁴C is 5730 years. What is the activity after 11,460 years?

Compute the number of half-lives: \[ n = \frac{t}{T_{1/2}} = \frac{11460}{5730} = 2 \]
Fraction remaining after 2 half-lives: \[ \left(\frac{1}{2}\right)^2 = \frac{1}{4} = 0.25 \]
Final activity: \[ A(t) = A_0 \times 0.25 = 1000 \,\text{Bq} \times 0.25 = 250 \,\text{Bq} \]

If you enter these values into the calculator (A₀ = 1000 Bq, T½ = 5730 years, t = 11,460 years), it will return a final activity of 250 Bq, 25% remaining, and 75% decayed.

Common half-lives

Here are approximate half-lives of some commonly discussed radionuclides:

Carbon‑14 (¹⁴C): 5730 years
Iodine‑131 (¹³¹I): 8.02 days
Cobalt‑60 (⁶⁰Co): 5.27 years
Uranium‑238 (²³⁸U): 4.47 × 10⁹ years
Radium‑226 (²²⁶Ra): 1600 years

Always consult an up‑to‑date nuclear data table for precise values when doing safety‑critical or regulatory work.

Safety note

This calculator is for educational and planning purposes only. It does not replace professional radiological protection advice, regulatory guidance, or instrument measurements. For any work involving ionising radiation, follow local regulations and consult a qualified health physicist or radiation safety officer.

FAQ

What is radioactive decay?

Radioactive decay is the spontaneous transformation of an unstable atomic nucleus into a more stable configuration, usually accompanied by the emission of particles (alpha, beta) or electromagnetic radiation (gamma). The process is probabilistic for individual atoms but predictable in bulk via the decay law.

What is the difference between activity, atoms, and mass in the decay law?

Activity \(A\) is the number of decays per second (measured in becquerels, Bq). It is proportional to the number of undecayed atoms \(N\), which in turn is proportional to the mass \(m\) of the radionuclide. Because of this proportionality, all three follow the same exponential decay law and the same half-life.

How many half-lives does it take for a sample to be “safe”?

There is no universal number of half-lives that guarantees safety. A common rule of thumb is that after about 10 half-lives, less than 0.1% of the original activity remains. Whether that is safe depends on the initial activity, type of radiation, exposure pathway, and regulatory limits.

Can this calculator handle growth as well as decay?

No. This tool assumes pure exponential decay with a single decay constant. More complex situations such as decay chains, ingrowth of daughter products, or continuous production (e.g., activation) require solving coupled differential equations and are beyond the scope of this simple calculator.

What units should I use for time and half-life?

You can choose any consistent time units (seconds, minutes, hours, days, years). The calculator automatically converts between your chosen half-life unit and time unit so that the decay constant is computed correctly.

Radioactive Decay Calculator