Radioactive Activity Calculator & Decay Explained

This calculator works out the radioactive activity (in becquerels or curies) of a sample from its initial activity, elapsed time, and half-life, or it can find the remaining fraction after a given number of half-lives. You enter the starting activity, the time that has passed, and the isotope’s half-life, and you get the current activity plus the fraction remaining. It uses the standard exponential decay law, so you can check a textbook problem or plan a lab experiment.

Formula

The activity A after time t is given by:

A = A₀ × 0.5^{t / T_½}

where:

• A₀ = initial activity (becquerels, Bq, or curies, Ci)
• t = elapsed time (same unit as half-life)
• T_½ = half-life (same unit as time)

The fraction remaining is simply 0.5^t/T_½. If you prefer the decay constant form, use A = A₀ e^−λt, where λ = ln(2) / T_½.

Worked Example

Using the default values: initial activity 1000 Bq, elapsed time 10 hours, half-life 6 hours.

1. Calculate the exponent: t / T_½ = 10 / 6 = 1.6667
2. Raise 0.5 to that power: 0.5^1.6667 = 0.31498
3. Multiply by initial activity: 1000 × 0.31498 = 314.98 Bq

So after 10 hours, the activity is about 315 Bq, and the fraction remaining is 0.315 (31.5%).

What Radioactive Activity Means

Activity is the number of nuclear disintegrations per second in a sample. The SI unit is the becquerel (Bq), where 1 Bq = 1 decay per second. The older unit is the curie (Ci), where 1 Ci = 3.7 × 10¹⁰ Bq (originally the activity of 1 gram of radium-226). Activity is not the same as dose or exposure—it tells you how many atoms are decaying, not how much energy is deposited in tissue.

Every radioactive isotope has a characteristic half-life, the time for half of the atoms to decay. Half-lives range from fractions of a second (polonium-214: 164 microseconds) to billions of years (uranium-238: 4.5 billion years). The decay law is probabilistic for a single atom, but for a large sample the rate is extremely predictable.

How to Choose Your Inputs

To get a meaningful result, you need three things:

• Initial activity – often printed on a vial or source, or measured with a detector. If you only know the number of atoms (N), multiply by the decay constant λ to get activity: A = λN.
• Elapsed time – the time between the initial measurement and now. Make sure it is in the same unit as the half-life (hours, days, years).
• Half-life – a fixed property of the isotope. Look it up from a reliable table (e.g., from a national laboratory or textbook).

If you want to find the time for activity to drop to a certain level, rearrange the formula: t = T_½ × log₂(A₀/A).

Common Mistakes and Misconceptions

• Mixing units – if half-life is in days but time is in hours, the exponent is wrong. Convert first.
• Confusing activity with dose – activity in Bq does not tell you the biological effect. Dose depends on energy, type of radiation, and tissue.
• Assuming linear decay – activity drops exponentially, not by a fixed amount per time. After two half-lives, it is 25%, not 0%.
• Rounding too early – if you round the exponent to 1.67, you get 314.3 Bq instead of 315.0 Bq. Keep extra digits until the end.

A common question: “If half-life is 6 hours, why isn’t it zero after 12 hours?” Because each half-life reduces the remaining amount by half, so after 12 hours you have 25% left, after 18 hours 12.5%, and so on—it never truly reaches zero.

Limits of the Formula

The exponential decay law assumes a pure isotope with a constant half-life and no production of new atoms (e.g., from a chain decay or neutron activation). In reality, many samples contain mixtures, and decay chains can cause the activity of a daughter product to rise temporarily. Also, the formula works for large numbers of atoms; for a handful of atoms, statistical fluctuations matter.

For very long-lived isotopes, the activity changes so slowly that you might not notice any decrease over a human lifetime. For example, uranium-238 has a half-life of 4.5 billion years, so a 1 kg sample has an activity of about 12,400 Bq—it decays, but the change is imperceptible on human timescales.

Practical Uses

This calculation appears in many fields:

• Medicine – estimating how much of a radiopharmaceutical remains in a patient after a known time.
• Research – correcting for decay when measuring a sample later than the calibration date.
• Environmental monitoring – tracking the decay of fallout or contamination.
• Education – solving textbook problems on nuclear physics.

For safety, always follow local regulations and use proper shielding and handling procedures. The calculator gives you a number, but it does not tell you the risk—that depends on exposure pathways and limits set by authorities.

Frequently Asked Questions

What is the difference between Bq and Ci?

Both measure activity. 1 becquerel is one decay per second; 1 curie is 3.7 × 10¹⁰ decays per second (37 GBq). Curies are still used in the United States, while becquerels are the international standard.

How do I find the half-life of an isotope?

Half-lives are measured experimentally and listed in standard references such as the Table of Nuclides from the International Atomic Energy Agency or the CRC Handbook. You can also search online for “half-life [element]” but verify the source.

Can I use this for a mixture of isotopes?

No, the formula assumes a single isotope. For a mixture, you would need to add the activities of each isotope, each decaying with its own half-life.

What if I only know the number of atoms, not the activity?

Multiply the number of atoms by the decay constant λ = ln(2)/T_½. For example, 10¹² atoms of an isotope with a half-life of 1 hour gives λ = 0.693 h⁻¹, so activity = 0.693 × 10¹² decays per hour, which is about 1.93 × 10⁸ Bq.

Does activity decrease linearly with time?

No, it decreases exponentially. After one half-life you have 50%, after two 25%, after three 12.5%, etc. The decay rate itself decreases because there are fewer atoms left to decay.

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in a0 | Initial activity | Bq | 1000 | 0
in t | Elapsed time | h | 10 | 0
in th | Half-life | h | 6 | 0
out a | Current activity | Bq | a0 * 0.5^(t/th) | 4
out frac | Fraction remaining | - | 0.5^(t/th) | 4