Radioactive Decay Calculator: Half-Life & Remaining Amount

What Is a Radioactive Decay Calculator?

A radioactive decay calculator finds how much of a radioactive substance is left after a given amount of time, based on its half-life. Enter a starting quantity, a half-life, and an elapsed time, and the calculator returns the remaining amount along with a chart of the decay curve.

Radioactive decay is a natural process in which unstable atomic nuclei release energy as radiation and turn into a different, more stable nucleus. Each radioactive isotope decays at its own fixed rate, described by its half-life: the time it takes for half of the atoms in a sample to decay. This rate does not change with temperature, pressure, or the chemical form of the substance, which is why decay calculations are reliable enough for uses ranging from cancer treatment to dating ancient artifacts.

Radioactive Decay Formula

The calculator uses the standard exponential decay equation:

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

• $N(t)$ — quantity remaining after time $t$
• $N_0$ — starting quantity
• $t$ — elapsed time
• $t_{1/2}$ — half-life of the isotope

The exponent $t / t_{1/2}$ counts how many half-lives have passed. After one half-life, half the original amount remains. After two half-lives, a quarter remains. After three, an eighth remains, and so on.

The same law is often written with a decay constant, $\lambda$, instead of half-life:

$N(t) = N_0 \times e^{-\lambda t}, \qquad \lambda = \frac{\ln 2}{t_{1/2}} \approx \frac{0.693}{t_{1/2}}$

Both forms give identical results. The half-life form is easier to reason about by hand, since $(1/2)^n$ has an obvious meaning for a whole number of half-lives.

Understanding Half-Life

Half-lives vary enormously between isotopes. Polonium-214 has a half-life of about 164 microseconds. Tellurium-128 has a half-life of roughly 2.2 septillion (2.2 × 10²⁴) years. After about 10 half-lives, less than 0.1% of a sample remains, which is why 10 half-lives is often used as a rough rule of thumb for when a radioactive source is "effectively gone."

How to Calculate Radioactive Decay

1. Enter the initial quantity. This can be in any unit — grams, atoms, becquerels, curies — and the result is given in the same unit.
2. Enter the half-life of the isotope and its time unit (seconds, minutes, hours, days, or years).
3. Enter the elapsed time and its unit. The unit does not need to match the half-life's unit; the calculator converts both to the same unit before computing the result.
4. Read the remaining quantity and the decay curve chart, which shows the amount over the full time period entered.

Worked Example

Iodine-131, used in thyroid treatment and imaging, has a half-life of 8.02 days. Starting with 1,000 millicuries, how much remains after 24.06 days?

$t / t_{1/2} = 24.06 / 8.02 = 3$ $N(t) = 1000 \times (1/2)^3 = 1000 \times 0.125 = 125 \text{ millicuries}$

Three half-lives have passed, so the sample has been halved three times: 1,000 → 500 → 250 → 125 millicuries.

Common Isotopes and Their Half-Lives

Real-World Uses of Decay Calculations

Medicine. Radiation therapy sources such as cobalt-60 lose strength as they decay, so exposure times must be increased over the life of the source to deliver the same dose. Nuclear medicine scans use short-lived isotopes like technetium-99m, timed so the scan happens while activity is high enough to image but low enough to limit the patient's radiation dose. Because its half-life is 6 hours, after 24 hours (four half-lives) about 6% of the original activity remains; after two days (eight half-lives) less than 1% remains.

Dating. Archaeologists estimate the age of organic remains by measuring how much carbon-14 is left, using its 5,730-year half-life. This method works up to roughly 50,000–60,000 years, beyond which too little carbon-14 remains to measure. Geologists date rocks over much longer spans using uranium-lead and potassium-argon decay, which is how Earth's age was estimated at about 4.54 billion years.

Safety and industry. After a nuclear accident, short-lived isotopes like iodine-131 pose an immediate but temporary hazard, while long-lived ones like cesium-137 (about 30 years) remain a concern for decades. Industrial radiography sources are retired once decay makes exposure times impractically long, and spent nuclear fuel is stored in cooling pools for years because it keeps generating heat as it decays.

Limitations of This Calculator

This calculator handles simple decay of a single isotope into a stable product. It does not model:

• Decay chains. Many heavy isotopes decay through a series of radioactive daughter products before reaching a stable one. Uranium-238, for example, passes through about 14 steps before becoming stable lead-206. Working out the quantity of each intermediate isotope requires solving the Bateman equations, a set of coupled differential equations.
• Small samples. The formula describes the average behavior of large numbers of atoms. With only a few dozen atoms, the actual decay count in any interval fluctuates around the predicted curve.
• Calibration. For dating applications, raw calculated ages are not the same as calendar years. Carbon dating, for instance, needs calibration curves that correct for past changes in atmospheric carbon-14.

Frequently Asked Questions

How is half-life defined? Half-life is the time it takes for half of the radioactive atoms in a sample to decay. It is constant for a given isotope regardless of sample size or physical conditions.

Can radioactive decay be sped up or slowed down? Under normal conditions, no. Decay rates are unaffected by temperature, pressure, or chemical bonding. This constancy is what makes radiometric dating and dose planning reliable. A few exotic decay modes, such as electron capture, show tiny rate changes only under extreme conditions like the interior of a star, which do not apply to everyday use.

What happens after many half-lives have passed? The remaining amount keeps halving but never mathematically reaches zero. After 10 half-lives, about 0.0977% remains. After 20 half-lives, less than one-millionth of the original amount remains. In radiation safety, 10 half-lives is a common threshold for treating a source as no longer significant.

How do I convert between time units? 1 year = 365.25 days, 1 day = 24 hours, 1 hour = 60 minutes, 1 minute = 60 seconds. The calculator converts automatically, so the half-life and elapsed time can be entered in different units.

Can this calculator be used for carbon dating? It can give a basic estimate using carbon-14's 5,730-year half-life. Professional dating uses published calibration curves to convert the raw result into calendar years, since atmospheric carbon-14 levels have varied over time.

Does this calculator handle decay chains? No. It calculates single-step decay from a parent isotope to a stable product. Isotopes that decay through a chain of radioactive daughters, such as uranium-238, require the Bateman equations, which this tool does not solve.