Half-Life Calculator: How to Calculate Radioactive Decay (With Examples)
Whether you’re dating archaeological samples with carbon-14, planning a nuclear medicine scan, or studying exponential decay in a chemistry class, half-life is the number that ties everything together.
The math is straightforward — but rearranging the decay equation by hand, converting time units, and keeping track of logarithms is easy to get wrong when you’re in a hurry.
The Tooladex Half-Life Calculator solves for remaining amount, initial amount, half-life, or elapsed time instantly. It also shows percent remaining, half-lives elapsed, and the decay constant — with presets for common isotopes like Carbon-14 and Technetium-99m.
Here’s what half-life means, how the formulas work, and how to use the calculator effectively.
☢️ What Is Half-Life?
Half-life (t½) is the time required for half of a radioactive substance (or any quantity undergoing first-order exponential decay) to disappear.
Key ideas:
- After 1 half-life, 50% remains
- After 2 half-lives, 25% remains
- After 3 half-lives, 12.5% remains
- After n half-lives, the fraction remaining is (½)ⁿ
Half-life is constant for a given isotope or decay process. It does not depend on how much material you start with — whether you have 1 gram or 1 kilogram of Carbon-14, the half-life is always about 5,730 years.
The same concept applies beyond nuclear physics:
- Radiocarbon dating (Carbon-14)
- Nuclear medicine (short-lived imaging isotopes)
- Pharmacokinetics (drug elimination half-life)
- Radiation safety (how long until activity drops to a safe level)
🧮 The Half-Life Formula
The standard decay equation is:
N = N₀ × (½)^(t/t½)
Where:
- N = amount remaining after time t
- N₀ = initial amount
- t = elapsed time
- t½ = half-life
Equivalently, using the decay constant λ:
N = N₀ × e^(-λt)
with λ = ln(2) / t½.
The number of half-lives elapsed is:
n = t / t½
🔢 How to Solve for Each Variable
Find Remaining Amount (N)
Formula: N = N₀ × (½)^(t/t½)
Example: You start with 800 Bq of an isotope. After 3 half-lives, how much remains?
- n = 3, so N = 800 × (½)³ = 800 × 0.125 = 100 Bq
Find Initial Amount (N₀)
Formula: N₀ = N × 2^(t/t½)
Example: 50 mg remains after 20 years. The half-life is 10 years. What was the initial amount?
- t/t½ = 20/10 = 2 half-lives
- N₀ = 50 × 2² = 200 mg
Find Elapsed Time (t)
Formula: t = t½ × log₂(N₀/N)
Example: A sample drops from 1,000 counts to 125 counts. Half-life is 6 hours. How much time passed?
- N₀/N = 1000/125 = 8 = 2³ → 3 half-lives
- t = 6 × 3 = 18 hours
Find Half-Life (t½)
Formula: t½ = t × ln(2) / ln(N₀/N)
Example: A substance goes from 400 units to 100 units in 12 days. What is the half-life?
- N₀/N = 4 = 2² → 2 half-lives in 12 days
- t½ = 12 / 2 = 6 days
📊 Common Isotope Half-Lives
| Isotope | Half-Life | Typical Use |
|---|---|---|
| Carbon-14 (¹⁴C) | 5,730 years | Radiocarbon dating |
| Uranium-238 (²³⁸U) | 4.47 billion years | Geological dating |
| Iodine-131 (¹³¹I) | 8.02 days | Thyroid treatment |
| Technetium-99m (⁹⁹ᵐTc) | 6 hours | Medical imaging |
| Radium-226 (²²⁶Ra) | 1,600 years | Historical sources |
| Tritium (³H) | 12.32 years | Radiolabeling, dating |
Our calculator includes one-click presets for these isotopes so you don’t have to look up half-lives manually.
🌍 Real-World Examples
Example 1: Carbon-14 Dating
An organic sample today has 25% of its original Carbon-14 compared to when it died. Carbon-14’s half-life is 5,730 years.
How old is the sample?
- 25% remaining = 2 half-lives (100% → 50% → 25%)
- Age = 2 × 5,730 = 11,460 years
Example 2: Technetium-99m in Nuclear Medicine
A clinic prepares 400 MBq of Tc-99m (half-life 6 hours) at 8:00 AM. How much activity remains at 8:00 PM the same day?
- Elapsed time = 12 hours = 2 half-lives
- N = 400 × (½)² = 100 MBq
Short half-lives are ideal for imaging — the tracer decays quickly so radiation exposure stays low.
Example 3: Drug Elimination
A medication has an elimination half-life of 4 hours. If plasma concentration starts at 80 mg/L, what concentration remains after 12 hours?
- 12 / 4 = 3 half-lives
- 80 × (½)³ = 10 mg/L
Pharmacists use the same math as nuclear decay when clearance follows first-order kinetics.
🛠️ How to Use the Tooladex Half-Life Calculator
- Choose what to solve for — remaining amount, initial amount, half-life, or elapsed time
- Pick a time unit — seconds, minutes, hours, days, or years (use the same unit for t and t½)
- Enter the three known values — results update automatically
- Optional: click an isotope preset to fill in a known half-life
The calculator also displays:
- Percent and fraction remaining
- Number of half-lives elapsed
- Decay constant λ (per second)
- A visual progress bar showing how much remains
❓ Frequently Asked Questions
What is the half-life formula?
N = N₀ × (½)^(t/t½). You can also write N = N₀ × e^(-λt) where λ = ln(2)/t½.
Why is half-life constant?
For first-order decay, the fraction lost per unit time stays the same regardless of how much is left. That’s why each isotope has a fixed half-life.
How many half-lives until a sample is “gone”?
Decay never reaches exactly zero in theory. In practice, after about 10 half-lives less than 0.1% remains — often treated as negligible.
Can half-life apply to drugs?
Yes. Many drugs are described by an elimination half-life — the time for plasma concentration to drop by half.
Do I need the same time units for t and t½?
Yes. If half-life is in years, elapsed time must also be in years. The calculator lets you pick one unit for both.
🎓 Conclusion
Half-life is the bridge between exponential decay and practical questions: How much is left? How long ago was this? When will it be safe?
With the Tooladex Half-Life Calculator, you can:
- Solve for any variable in the decay equation
- Work in seconds through years with one unit selector
- Load common isotope half-lives instantly
- See percent remaining, half-lives elapsed, and λ at a glance
Whether you’re in a chemistry lab, a physics lecture, or planning around a medical isotope, the math should take seconds — not a spreadsheet.
Try it now: enter your values and get the answer instantly.
Half-Life Calculator
Calculate radioactive decay using half-life. Solve for remaining amount, initial amount, half-life, or elapsed time with decay constant, percent remaining, and common isotope presets.