Use this half-life calculator to find how much of a substance remains after any amount of time. Enter a starting quantity, the known half-life, and elapsed time. The decay calculator returns the remaining amount instantly.
Half-life is the amount of time it takes for a quantity to decrease to half its starting value. It applies to radioactive isotopes in nuclear physics, drugs eliminated from the body, and chemical reactions in chemistry. The same exponential decay formula drives every calculation on this page.
How the Half-Life Calculator Works
The calculator uses three inputs:
- Initial quantity (the amount you start with)
- Half-life (how long one half-life lasts, in your chosen time unit)
- Elapsed time (total time that has passed)
From those values it solves the exponential decay formula and returns the remaining quantity. You can also reverse the calculation: enter a starting and ending amount to find the elapsed time or the half-life itself.
Do units matter? Yes, but only for consistency. Seconds, minutes, hours, days, or years all work as long as the half-life and elapsed time share the same unit. The calculator handles the conversion math once you pick a unit.
Half-Life Formula and Equation
The core half-life equation is:
N(t) = N₀ × (1/2)^(t / t½)
Where:
- N(t) is the quantity remaining after time t
- N₀ is the initial quantity
- t is the elapsed time
- t½ is the half-life
This formula assumes first-order exponential decay. It works for radioactive decay, most drug elimination, and many chemical processes.
Decay Constant and Mean Lifetime in the Formula
Two related values appear in many textbooks:
- Decay constant (λ): the probability of decay per unit time. λ = ln(2) / t½, which is approximately 0.693 / t½.
- Mean lifetime (τ): the average time a single atom or molecule survives before decaying. τ = 1 / λ, or equivalently t½ / ln(2).
You can rewrite the equation using the decay constant:
N(t) = N₀ × e^(−λt)
Both forms produce the same result. The calculator uses whichever version fits the inputs you provide.
How to Calculate Half-Life Rate from a Substance
If you know a starting amount and the amount remaining after a measured interval of time, you can solve for the half-life:
- Divide the remaining quantity by the initial quantity.
- Take the natural log of that ratio.
- Divide by the elapsed time to get −λ.
- Convert: t½ = ln(2) / λ.
Example: A 100 g sample drops to 25 g in 10 days.
- Ratio: 25 / 100 = 0.25
- ln(0.25) = −1.386
- λ = 1.386 / 10 = 0.1386 per day
- t½ = 0.693 / 0.1386 ≈ 5 days
Two half-lives passed (100 → 50 → 25), which checks out: 2 × 5 = 10 days.
Radioactive Decay Calculator
Radioactive decay follows the half-life formula precisely. Every radioactive isotope has a fixed half-life that does not change with temperature, pressure, or chemical environment.
Enter the isotope's known half-life and your sample size. The calculator returns how much remains after any date and time you choose.
Decay of a Substance Over Date and Time
To track decay across a calendar span, convert the date range into a single time unit, then apply the formula.
Example: You have 60 grams of Np-240 (half-life ≈ 1 hour). How much remains after 4 hours?
- Number of half-lives: 4 / 1 = 4
- Remaining: 60 × (1/2)⁴ = 60 × 0.0625 = 3.75 grams
After 4 half-lives, only about 6.25% of the original sample remains.
Radiocarbon Dating and Isotope Half-Life Example
Carbon-14 has a half-life of approximately 5,730 years. Scientists measure the ratio of C-14 remaining in organic material to estimate its age. This is the basis of radiocarbon dating.
Example: A plant sample contains 25% of its original C-14.
- 25% means two half-lives have passed (100% → 50% → 25%).
- Estimated age: 2 × 5,730 = approximately 11,460 years.
The calculator performs this math for any isotope when you enter the correct half-life value.
List of Radionuclides with Long Half-Lives
Some isotopes decay so slowly they persist in the environment for thousands or billions of years.
| Isotope | Half-Life (approx.) | Common Context |
|---|---|---|
| Uranium-238 | 4.5 billion years | Geology, nuclear fuel |
| Potassium-40 | 1.25 billion years | Rock dating, natural background |
| Carbon-14 | 5,730 years | Radiocarbon dating |
| Plutonium-239 | 24,100 years | Nuclear waste |
| Cesium-137 | 30.2 years | Environmental monitoring |
| Cobalt-60 | 5.27 years | Medical, industrial |
| Actinium-225 | 10 days | Targeted cancer therapy |
After 720 hours (30 days), approximately 87.5% of an Ac-225 sample has decayed, leaving roughly 12.5% of the original amount (about 3 half-lives).
If your isotope is not listed, enter its published half-life directly into the calculator.
Drug Half-Life Calculator
The same exponential decay math applies to how the body eliminates a drug. After administration, the drug's concentration in the bloodstream drops by half with each half-life period.
This section provides estimates for educational planning. It is not medical advice. Always consult a healthcare professional about dosing, interactions, and elimination.
Half-Life of a Drug in Medicine
In pharmacology, the half-life of a drug tells clinicians how long the medication stays active. It influences:
- Dosing frequency: shorter half-life drugs need more frequent doses.
- Time to steady state: it takes roughly 4 to 5 half-lives of repeated dosing to reach a stable concentration.
- Elimination: after 5 half-lives, approximately 97% of a single dose has been eliminated.
This is often called the 5 half-life rule. After five half-lives, the remaining amount (about 3%) is generally considered negligible.
What is a half-life of 4 hours? It means the drug's concentration drops by half every 4 hours. After 20 hours (5 half-lives), roughly 97% of the dose is gone.
How to Calculate Drug Half-Life with an Example
Example: A patient takes 200 mg of a drug with a half-life of 6 hours. How much remains after 18 hours?
- Number of half-lives: 18 / 6 = 3
- Remaining: 200 × (1/2)³ = 200 × 0.125 = 25 mg
This assumes first-order elimination, which describes most drugs at therapeutic doses. Some drugs follow more complex kinetics, so real-world elimination may differ.
Does this calculator assume first-order behavior? Yes. The formula models exponential (first-order) decay. For drugs with zero-order or mixed kinetics (like alcohol or phenytoin at high doses), this estimate will be less accurate.
List of Drug Half-Lives and Source Data
Below are approximate half-lives for commonly searched medications. Values can vary based on individual factors like age, liver function, kidney function, and genetics.
| Drug | Approximate Half-Life | Source |
|---|---|---|
| Adderall (amphetamine) | 10 to 13 hours | FDA prescribing information |
| Ibuprofen | 2 to 4 hours | FDA prescribing information |
| Acetaminophen | 2 to 3 hours | FDA prescribing information |
| Amoxicillin | 1 to 1.5 hours | FDA prescribing information |
| Diazepam (Valium) | 20 to 100 hours | FDA prescribing information |
| Metformin | 4 to 8.7 hours | FDA prescribing information |
| Fluoxetine (Prozac) | 1 to 3 days (active metabolite: 4 to 16 days) | FDA prescribing information |
Diazepam is among the drugs with the longest commonly cited half-lives. Ibuprofen and amoxicillin are examples of drugs with a short half-life.
These values are estimates from published prescribing data. Your actual elimination rate depends on personal health factors. Consult your physician or pharmacist for guidance specific to your situation.
Decay Calculator Example: Step by Step
Let's walk through a complete calculation using the formula.
Problem: How long will it take for 18.0 grams of Ra-226 to decay to 2.25 grams? Ra-226 has a half-life of 1,600 years.
Step 1: Find how many half-lives are needed.
- 18.0 → 9.0 (1 half-life)
- 9.0 → 4.5 (2 half-lives)
- 4.5 → 2.25 (3 half-lives)
Three half-lives reduce 18.0 g to 2.25 g.
Step 2: Calculate total time.
- 3 × 1,600 = 4,800 years
Step 3: Verify with the formula.
- N(t) = 18.0 × (1/2)^(4800/1600) = 18.0 × (1/2)³ = 18.0 × 0.125 = 2.25 g ✓
The answer checks out. Enter these same values into the calculator above to confirm.
Formula Estimate Limitations for This Calculator
This calculator provides estimates based on the standard exponential decay formula. Keep these limitations in mind:
- Radioactive decay follows the formula very precisely for large sample sizes. For very small numbers of atoms, statistical variation increases.
- Drug elimination in real patients is influenced by liver function, kidney function, age, body composition, drug interactions, and genetics. The calculator assumes a single fixed half-life with first-order kinetics.
- Chemical reactions may not follow pure first-order decay. Verify the reaction order before using this tool.
- The calculator does not account for repeated dosing, drug accumulation, or active metabolites.
This tool is for educational and planning purposes. It is not a substitute for professional medical, scientific, or environmental analysis. For drug dosing decisions, consult a qualified healthcare professional. For precise isotope measurements, consult a radiation safety officer or accredited laboratory.