Introduction
Radioactive atoms are unstable. Over time, they break apart and release energy in a process called radioactive decay. Each radioactive isotope decays at its own fixed rate, measured by its half-life — the time it takes for half of the atoms in a sample to decay. This is a key idea in nuclear physics and is used in medicine, energy, archaeology, and environmental science.
This Radioactive Decay Calculator helps you find how much of a radioactive substance remains after a given period of time. Enter the isotope, its half-life, the starting activity, and the time that has passed, and the tool does the rest. It uses the standard decay formula N(t) = N₀ × (½)^(t/t½) to give you the remaining activity, the decayed amount, the percent remaining, and the number of half-lives elapsed. It also shows the decay constant (λ) and mean lifetime (τ) for your isotope. A built-in chart and decay schedule table let you see how the activity drops over time.
The calculator includes a database of over 250 common radioactive isotopes so you can quickly select one, or you can type in a custom half-life for any isotope not listed. You can also use date-based input to find the activity between two specific dates, or run a back-decay calculation to figure out what the original activity was. Additional tools on this page let you solve the half-life formula for any unknown variable and convert between half-life, mean lifetime, and decay constant. For a simpler calculation focused solely on half-life, you may also find our Half Life Calculator useful.
How to Use Our Radioactive Decay Calculator
Enter information about a radioactive isotope, its starting activity, and a time period to find out how much of the substance remains, how much has decayed, and other key values like the decay constant and mean lifetime. This tool also includes a half-life formula solver and a converter for switching between half-life, mean lifetime, and decay constant.
Radioactive Decay Calculator
Element: Pick an element from the dropdown list. The calculator has a built-in database of common radioactive elements sorted alphabetically, from Actinium to Zirconium.
Isotope: Once you pick an element, choose the specific isotope you want to work with. The half-life will fill in automatically based on your selection.
Half-Life: This field shows the known half-life of the selected isotope. You can change the value or the time unit (seconds, minutes, hours, days, months, or years) if needed.
Isotope Not Listed: Check this box if your isotope is not in the database. You can then type in the isotope name and its half-life by hand.
Initial Activity (N₀): Enter the starting activity or quantity of your radioactive sample. Choose a unit from the dropdown, including Becquerels (Bq, kBq, MBq, GBq, TBq), Curies (Ci, mCi, μCi, nCi), disintegrations per minute (dpm), counts per minute (cpm), atoms, or grams.
Time Period — Elapsed Time: Select this option to enter the amount of time that has passed. Fill in the years, days, hours, and minutes fields as needed.
Time Period — Date-Based: Select this option to use specific dates instead. Enter the original reference date and time and the target measurement date and time, and the calculator will figure out how much time has passed.
Back-Decay: Check this box if you want to work backward in time. This lets you calculate what the original activity was based on a current measurement.
After you press Calculate Decay, the tool shows the remaining activity, decayed amount, percent remaining, number of half-lives elapsed, decay constant (λ), and mean lifetime (τ). It also draws a decay curve chart and builds a decay schedule table.
Half-Life Formula Solver
Solve For: Use this dropdown to pick which variable you want to find. You can solve for the remaining quantity N(t), the initial quantity N₀, the time elapsed (t), or the half-life (t½).
Initial Quantity (N₀): Enter the starting amount of the substance. This field is hidden when you are solving for N₀.
Remaining Quantity N(t): Enter the amount left after decay. This field is hidden when you are solving for N(t).
Time Elapsed (t): Enter how much time has passed and pick a time unit. This field is hidden when you are solving for time.
Half-Life (t½): Enter the half-life value and pick a time unit. This field is hidden when you are solving for half-life.
Press Solve and the calculator displays the answer using the standard decay formula N(t) = N₀ × (½)^(t / t½).
Half-Life / Mean Lifetime / Decay Constant Converter
Half-Life (t½): Enter a half-life value and choose the time unit. This is the time it takes for half of a radioactive sample to decay.
Mean Lifetime (τ): Enter a mean lifetime value and choose the time unit. The mean lifetime is the average time a single atom survives before decaying.
Decay Constant (λ): Enter a decay constant value and choose the rate unit (per second, minute, hour, day, or year). The decay constant tells you the probability of decay per unit of time.
Convert From: Select which value you are entering — Half-Life, Mean Lifetime, or Decay Constant — and press Convert. The other two values will be calculated automatically using the relationships λ = ln(2) / t½ and τ = 1 / λ.
What Is Radioactive Decay?
Radioactive decay is the process by which an unstable atomic nucleus loses energy by releasing radiation. Over time, a radioactive substance breaks down into a more stable form. This happens naturally and cannot be sped up or slowed down by changes in temperature, pressure, or chemistry. Every radioactive isotope decays at its own fixed rate, which scientists describe using a value called the half-life. The exponential nature of this process is closely related to other physics concepts — for instance, a Free Fall Calculator also relies on time-dependent equations to describe how systems change, though governed by gravity rather than nuclear instability.
Understanding Half-Life
The half-life of a radioactive isotope is the amount of time it takes for exactly half of the atoms in a sample to decay. For example, Cesium-137 has a half-life of about 30.17 years. If you start with 1,000 units of Cs-137, after 30.17 years you will have roughly 500 units left. After another 30.17 years, you will have about 250 units, and so on. The substance never fully disappears in theory, but it gets smaller and smaller with each half-life that passes. Our dedicated Half Life Calculator can help you quickly determine how many half-lives have elapsed for any given scenario.
Half-lives vary wildly between isotopes. Oxygen-15 has a half-life of just 122 seconds, while Uranium-238 takes about 4.47 billion years to lose half its atoms. This range is why different isotopes are useful for different purposes, from medical imaging to dating ancient rocks.
The Radioactive Decay Formula
The core equation used to calculate radioactive decay is:
N(t) = N₀ × (½)t/t½
In this formula, N₀ is the starting amount of the radioactive material, t is the time that has passed, t½ is the half-life, and N(t) is the amount remaining after time t. You can rearrange this equation to solve for any one of the four variables as long as you know the other three. Working with very large or very small numbers in these calculations often requires Scientific Notation Calculator skills, and understanding logarithms is essential when solving for time or half-life from the decay equation.
Decay Constant and Mean Lifetime
Two other values are closely related to half-life. The decay constant (λ) tells you the probability that a single atom will decay per unit of time. It is calculated as λ = ln(2) / t½, where ln(2) is approximately 0.693. The mean lifetime (τ) is the average time a single atom survives before decaying, calculated as τ = 1 / λ. Mean lifetime is always longer than the half-life by a factor of about 1.443.
Activity and Units of Measurement
The activity of a radioactive sample measures how many atoms decay per second. The SI unit of activity is the becquerel (Bq), which equals one decay per second. An older but still widely used unit is the curie (Ci), where 1 Ci equals 37 billion decays per second. Smaller units like millicuries (mCi), microcuries (μCi), and kilobecquerels (kBq) are common in medical and laboratory settings. Activity follows the same exponential decay pattern as the number of atoms, so the decay formula works equally well whether you measure in becquerels, curies, atom counts, or grams. If you need to determine the percentage of a sample that has decayed or remains, straightforward percentage math applies directly to the results.
Back-Decay: Working Backwards in Time
Sometimes you need to figure out how much radioactive material was present at an earlier date based on a current measurement. This is called back-decay. The same formula applies, but in reverse — you divide by the decay factor instead of multiplying. Back-decay is used in nuclear medicine to determine the original strength of a dose, and in forensic science to establish timelines.
Common Applications of Radioactive Decay Calculations
- Nuclear medicine: Doctors use isotopes like Technetium-99m (half-life of 6 hours) and Iodine-131 (half-life of 8 days) for imaging and treatment. Knowing the decay rate helps them plan the right dose at the right time.
- Radiocarbon dating: Scientists use the 5,730-year half-life of Carbon-14 to determine the age of organic materials up to about 50,000 years old.
- Nuclear waste management: Long-lived isotopes like Plutonium-239 (half-life of 24,110 years) require careful storage plans that account for thousands of years of slow decay.
- Radiation safety: Health physicists calculate how long it takes for a contaminated area to reach safe activity levels.
- Geology and Earth science: Uranium-lead dating uses the decay of U-238 (half-life of 4.47 billion years) to measure the age of rocks and the Earth itself.
Many of these applications intersect with other areas of physics. For example, understanding the energy released during nuclear decay connects directly to Einstein's famous mass–energy equivalence, which you can explore with our E = mc² Calculator. Radiation shielding calculations may involve understanding kinetic energy of emitted particles, while the Potential Energy Calculator can help illustrate the energy stored within atomic nuclei. When measuring experimental results, a Percent Error Calculator is invaluable for quantifying measurement uncertainty in decay rate experiments.
Key Takeaways
Radioactive decay is a predictable, exponential process governed by each isotope's unique half-life. Whether you are calculating how much of a medical isotope remains after a few hours, estimating the age of a fossil, or converting between half-life, mean lifetime, and decay constant, the underlying math stays the same. The formula N(t) = N₀ × (½)t/t½ is one of the most reliable equations in all of physics.