Activity Half-Life Calculator and Decay Formula

An activity half-life calculator estimates how much radioactive activity remains after a chosen time, or how long it takes activity to fall to a chosen level. Enter the starting activity, elapsed time, and half-life; the calculator returns the remaining activity and the fraction left.

Formula

Radioactive activity decreases exponentially. The remaining activity is A = A₀ × (1/2)^t/T. Here, A is remaining activity in becquerels (Bq), A₀ is initial activity in Bq, t is elapsed time, and T is the half-life. The calculator requires t and T to use the same time unit; its defaults use hours. The fraction remaining is f = (1/2)^t/T, a dimensionless value.

Activity is the rate of nuclear decays. One becquerel equals one decay per second. A curie (Ci) is another activity unit; 1 Ci equals 3.7 × 10¹⁰ Bq. The equation works with any consistent activity unit, so if you enter an initial value in another unit, interpret the result in that same unit.

Worked Example

Using the calculator’s defaults—initial activity 1,000 Bq, elapsed time 24 hours, and half-life 6 hours:

1. Find the number of half-lives: t/T = 24 h / 6 h = 4.
2. Find the remaining fraction: f = (1/2)⁴ = 1/16 = 0.0625.
3. Multiply by the starting activity: A = 1,000 Bq × 0.0625 = 62.5 Bq.

The calculator displays 62.50 Bq remaining and a remaining fraction of 0.0625. That fraction is 6.25% of the initial activity.

How to Use an Activity Half-Life Calculator

• Enter the activity at the start of the period, using a known unit such as Bq or Ci.
• Enter the elapsed time and the isotope’s half-life in the same unit. Convert days to hours, for example, if you enter the half-life in hours.
• Read the remaining activity and fraction. The fraction can be multiplied by 100 to express the result as a percentage.

To calculate time instead of remaining activity, rearrange the equation: t = T × log₂(A₀/A), where A is the target activity. This requires a target greater than zero and no greater than the starting activity. For instance, reaching one-eighth of the initial activity takes three half-lives, because 1/8 = (1/2)³.

What Half-Life Means

A radionuclide’s half-life is the time required for half of a large population of its unstable nuclei to decay. Because activity is proportional to the number of undecayed nuclei, activity follows the same half-life pattern under ordinary conditions. After one half-life, 50% remains; after two, 25%; after three, 12.5%.

Half-life is specific to the radionuclide, not to the size or shape of a sample. It is not a countdown for individual atoms: decay is random, and the equation describes the expected behavior of a population. For a small number of atoms, actual counts over a short interval can differ noticeably from the smooth mathematical curve.

Reading Results and Common Mistakes

The computed value is a mathematical estimate of activity at the specified elapsed time, assuming the starting activity and half-life are known and the sample behaves as a single radionuclide. A result of 62.5 Bq does not mean 62.5 atoms remain; it means an expected decay rate of 62.5 transformations per second at that time.

• Mixing time units: Do not divide 24 hours by a half-life of 6 days. Convert one value first; 6 days is 144 hours.
• Confusing activity with mass: The formula calculates activity, not remaining mass. Converting activity to mass requires additional information, including the radionuclide and its specific activity.
• Rounding too early: Keep full precision during intermediate steps and round only the displayed result. A rounded answer can be zero even though the model predicts a very small positive value.
• Using the wrong isotope: A half-life must correspond to the radionuclide being evaluated. A material containing several radionuclides may need a separate calculation for each component.

Limits and Related Calculations

This simple model assumes a constant, known half-life and a single radioactive component. It does not account for measurement uncertainty, environmental transport, shielding, chemical changes, daughter products, or a mixture of isotopes. In a decay chain, daughter nuclides may themselves be radioactive, so the total measured activity can behave differently from the decay of the parent alone.

Physical decay is also different from biological clearance or removal from a system. In contexts where material is both decaying and being removed, an effective half-life may be calculated from both processes, but that is a different model and requires suitable inputs. The calculator should not be treated as a measurement, a safety assessment, or a substitute for qualified radiation-safety guidance.

Frequently Asked Questions

How do I calculate remaining activity after several half-lives?
Divide elapsed time by the half-life to get the number of half-lives, then multiply the starting activity by (1/2) raised to that number. Four half-lives leave 1/16, or 6.25%, of the starting activity.

Can I use days or years instead of hours?
Yes. Any time unit works as long as elapsed time and half-life use the same unit. The activity unit remains unchanged by the time unit selected.

Does activity ever reach exactly zero?
The exponential formula approaches zero but remains positive for any finite elapsed time. In practice, a value may become too small to measure or may be rounded to zero.

Is half-life the time until all radioactive material is gone?
No. Each half-life reduces the expected amount by half; it does not eliminate the remainder. After five half-lives, 3.125% remains in the idealized model.

Can this calculator find the half-life?
The displayed calculator uses a known half-life to estimate remaining activity. If you know the initial and final activities and the elapsed time, rearrange the equation: T = t × log₂(A₀/A).

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in a0 | Initial activity | Bq | 1000 | 0
in t | Elapsed time | h | 24 | 0
in h | Half-life | h | 6 | 0
out a | Remaining activity | Bq | a0 * 0.5 ^ (t / h) | 2
out f | Fraction remaining | - | 0.5 ^ (t / h) | 4