You Have a Radioactive Sample and Need to Know How Much Is Left After 10 Years
Radioactive materials decay at a predictable rate. Each radioactive nucleus has a certain probability of decaying per unit time. For each isotope, the half-life is the time required for half of the sample to decay. Carbon-14 has a half-life of 5,730 years-this is used for dating archaeological artifacts. Uranium-235 has a half-life of 704 million years. Technetium-99m has a half-life of just 6 hours-this is used in medical imaging. This calculator computes how much radioactive material remains after any time period.
What This Calculator Does
This calculator applies the exponential decay formula to find the remaining quantity of a radioactive sample. You provide the initial quantity, half-life, and elapsed time, and it shows how much remains. It also works in reverse: if you measure a remaining quantity, it can estimate the age of the sample (useful in radiocarbon dating). The calculator handles any time units (years, days, hours, seconds) and any quantity units (grams, Becquerels, Curies).
How to Use This Calculator
Initial Quantity (N₀): Enter the starting amount of radioactive material in grams (g), kilograms (kg), Curies (Ci), Becquerels (Bq), or other units.
Half-Life (t½): Enter the half-life of the isotope in years, days, hours, seconds, or other time units. Look up this value for your specific isotope.
Elapsed Time (t): Enter how much time has passed since the initial measurement.
Remaining Quantity (N): The calculator shows how much material remains after the elapsed time.
The calculator also shows the decay constant (λ) and the decay rate (activity) in Becquerels or Curies.
The Formula Behind the Math
The amount of radioactive material remaining after time t follows exponential decay:
N(t) = N₀ × (1/2)^(t / t½)
Where:
Alternatively, using the decay constant λ = ln(2) / t½:
N(t) = N₀ × e^(−λt)
The activity (decay rate in Becquerels or Curies) is:
A(t) = λ × N(t) = λ × N₀ × e^(−λt)
Where 1 Becquerel (Bq) = 1 decay per second, and 1 Curie (Ci) = 3.7 × 10¹⁰ Bq.
Worked Example:
A medical sample contains 10 milligrams of Technetium-99m (t½ = 6 hours). How much remains after 24 hours?
After 24 hours, only 0.625 mg remains. This is why Tc-99m must be produced just before use-it decays too quickly for storage.
Now consider Carbon-14 dating. A bone sample contains 25% of the original C-14. How old is the sample?
The sample is approximately 11,460 years old (two half-lives).
Our calculator does all of this instantly, but now you understand exactly what it's computing.
Radiocarbon Dating
Living organisms continuously exchange carbon with the atmosphere, maintaining constant C-14. When they die, C-14 decay is no longer replenished. By measuring the remaining C-14, scientists determine when the organism died. The formula rearranges to:
t = t½ × log₂(N₀ / N)
This technique works for samples up to about 50,000 years old (10 half-lives, when the signal becomes too weak). Older samples have too little C-14 to measure reliably.
Nuclear Medicine
Radioactive isotopes are used as tracers in medical imaging. Technetium-99m (6-hour half-life) is injected into patients; a gamma camera detects where it concentrates. Its short half-life means it decays away quickly, minimizing radiation exposure. Iodine-131 (8-day half-life) is used to treat thyroid cancer; enough decays to destroy cancer cells while the body recovers from remaining radiation.
Nuclear Waste and Safety
Highly radioactive waste must be stored safely until it decays to harmless levels. Plutonium-239 has a half-life of 24,110 years-one of the longest. After 10 half-lives (241,110 years), it's down to 1/1024 of original activity. This is why nuclear waste repositories must remain secure for hundreds of thousands of years.
Radiological Accidents
After the Chernobyl disaster, Cs-137 (30-year half-life) and Sr-90 (29-year half-life) contaminated soil. After 1 half-life (30 years, around 2016), activity was down to 50%. After 2 half-lives (60 years, around 2046), it will be 25%. The region is still contaminated today, showing why long half-lives are so problematic.
Tips and Things to Watch Out For
Half-life and decay constant are related. λ = ln(2) / t½ ≈ 0.693 / t½. Longer half-lives mean smaller decay constants. Knowing one, you can always find the other.
Exponential decay never reaches zero. Mathematically, N(t) never becomes exactly zero, but it becomes so small that it's undetectable. After ~10 half-lives, the remaining amount is typically undetectable.
Units must be consistent. If half-life is in years, elapsed time must be in years. The calculator handles unit conversion, but verify the result makes sense.
Activity and quantity are different. Activity is the decay rate (Becquerels or Curies). Quantity is the total mass. A large quantity of long-lived isotope has low activity. A small quantity of short-lived isotope can have high activity. Both are important for safety calculations.
Mixing isotopes complicates calculations. A sample with multiple radioactive isotopes decays at different rates. Each isotope follows its own half-life. You must track each separately.
Frequently Asked Questions
What's the difference between half-life and decay constant?
Half-life (t½) is the time for 50% to decay. Decay constant (λ) is the probability per unit time of decay. They're related: λ = ln(2) / t½ ≈ 0.693 / t½. Half-life is more intuitive; decay constant appears in equations.
How is radiocarbon dating limited?
C-14 dating works for samples up to about 50,000 years old (roughly 10 half-lives). Beyond that, so little C-14 remains that measurement error dominates. Also, the assumption of constant atmospheric C-14 is violated for ancient samples (due to variations in Earth's magnetic field). Calibration curves account for this.
Why are some isotopes used in medicine and others aren't?
Medical isotopes must decay quickly (short half-life, like 6 hours for Tc-99m) so patients aren't exposed to long-term radiation. They must also emit gamma rays (detectable) rather than alpha or beta particles (blocked by skin). Isotopes like U-238 are too long-lived and don't emit gammas, so they're not useful medically.
How do I calculate the age of a sample from remaining activity?
Measure the current activity and the initial activity (or assume it equals modern activity for C-14). Use t = t½ × log₂(A₀ / A). This is the same formula as for quantity; activity decays at the same rate as quantity.
What's the difference between Curies and Becquerels?
Both measure activity (decays per second). 1 Becquerel (Bq) = 1 decay/second (SI unit). 1 Curie (Ci) = 3.7 × 10¹⁰ Bq ≈ 37 GBq (older unit, still used). 1 Millicurie (mCi) = 37 MBq. The Becquerel is standard now, but medical literature still often uses Curies.
Can you reverse radioactive decay?
No. Radioactive decay is spontaneous and irreversible (you can't un-decay a nucleus). However, different decay products (like neutron activation) can create new radioactive isotopes. But that's creating new radioactivity, not reversing existing decay.
Related Calculators
Use our Molar Mass Calculator to convert between mass and moles of isotopes. The Dilution Calculator helps prepare solutions of radioactive tracers at exact concentrations. For more physics concepts, explore our Density and Pressure Calculators.