Half-Life Calculator
Calculate radioactive decay using the half-life formula N(t) = N₀ × (½)^(t/t½). Enter any three of four variables and solve for the fourth. Also converts between half-life, mean lifetime, and decay constant.
What Is Half-Life? Definition and the Decay Formula
Half-life is the time required for a quantity undergoing exponential decay to reduce to exactly half of its initial value. In radioactive decay, it is the time for half the atoms in a sample to spontaneously disintegrate. The concept was formalised by Ernest Rutherford in 1907 and remains one of the most fundamental constants in nuclear physics, pharmacology, and environmental science.
The three equivalent mathematical expressions for exponential decay are:
1. Half-life form: N(t) = N₀ × (½)^(t / t½)
2. Mean lifetime form: N(t) = N₀ × e^(−t / τ)
3. Decay constant form: N(t) = N₀ × e^(−λt)
Where N₀ is the initial quantity, N(t) is the remaining quantity at time t, t½ is the half-life, τ (tau) is the mean lifetime, and λ (lambda) is the decay constant.
The Three Equivalent Measures of Decay Rate
Every radioactive isotope (or decaying substance) can be described by three equivalent constants that are mathematically interconvertible. Knowing any one gives you the other two:
| Measure | Symbol | Formula |
|---|---|---|
| Half-life | t½ | t½ = ln(2) / λ ≈ 0.6931 / λ |
| Mean lifetime | τ | τ = t½ / ln(2) ≈ 1.4427 × t½ |
| Decay constant | λ | λ = ln(2) / t½ ≈ 0.6931 / t½ |
The decay constant λ represents the probability per unit time that a single radioactive atom decays. A higher λ means faster decay and a shorter half-life.
The mean lifetime τ (also called the e-folding time) is the average time a single atom survives before decaying. It is always longer than the half-life because it is weighted towards atoms that survive longer.
Solving the Half-Life Equation for Any Variable
This calculator can solve for any one of the four variables when three are known:
| Solving for | Formula |
|---|---|
| Remaining quantity (Nₜ) | Nₜ = N₀ × (½)^(t/t½) |
| Initial quantity (N₀) | N₀ = Nₜ / (½)^(t/t½) |
| Time elapsed (t) | t = t½ × log(Nₜ/N₀) / log(½) |
| Half-life (t½) | t½ = t × log(½) / log(Nₜ/N₀) |
Real-World Applications of Half-Life Calculations
Radiocarbon Dating (Carbon-14)
Carbon-14 has a half-life of approximately 5,730 years. Living organisms continuously incorporate carbon-14 from the atmosphere. Once an organism dies, no new carbon-14 enters, and the existing amount decays at this predictable rate. By measuring the current ratio of carbon-14 to stable carbon-12, scientists can calculate time of death to approximately ±30–40 years for samples up to 50,000 years old.
Worked example: A fossil contains 25% of the carbon-14 present in a living sample.
- N(t)/N₀ = 0.25
- (½)^(t/5730) = 0.25
- t/5730 = log(0.25) / log(0.5) = 2
- t = 11,460 years
Nuclear Medicine
Technetium-99m (t½ = 6.02 hours) is the most widely used radioactive tracer in nuclear medicine. Its short half-life means it clears the body within 24 hours after imaging, minimising radiation exposure while providing clear scan resolution.
Iodine-131 (t½ = 8.02 days) is used to treat thyroid cancer and hyperthyroidism. The tumour absorbs the iodine while the short half-life limits systemic radiation damage.
Pharmacology and Drug Half-Life
Every drug has a biological half-life — the time for the body to eliminate half the dose. This determines dosing intervals:
- A drug with a 4-hour half-life requires dosing every 4–8 hours to maintain therapeutic concentration
- After approximately 5 half-lives (~97% eliminated), a drug is considered fully cleared from the body
- Aspirin: ~3–4 hours. Ibuprofen: ~2 hours. Fluoxetine (Prozac): ~1–4 days. Amiodarone: ~40–55 days
Environmental and Nuclear Safety
Strontium-90 (t½ ≈ 28.8 years) and caesium-137 (t½ ≈ 30.2 years) are both produced in nuclear fission and were released during the Chernobyl and Fukushima accidents. After 10 half-lives (~300 years), approximately 0.1% of the original amount remains — the standard threshold for considering an area effectively safe for habitation.
Frequently Asked Questions
What is the half-life formula?
N(t) = N₀ × (½)^(t/t½), where N₀ is the initial quantity, t is the elapsed time, and t½ is the half-life. Leave one of the four variables blank and the calculator solves for it.
What is the half-life of uranium-238?
Uranium-238 has a half-life of approximately 4.47 billion years — comparable to the age of Earth itself (4.54 billion years). This extreme stability is why uranium is still present in Earth's crust.
Does half-life change with temperature or pressure?
For nuclear decay, no — the half-life is virtually constant regardless of temperature, pressure, or chemical bonding. This is a key distinction from chemical reaction rates, which are highly condition-dependent. Some exotic conditions (extreme relativistic speeds causing time dilation, or electron capture in highly ionised atoms) can produce small variations.
What does it mean to say a drug has a half-life?
A drug's biological half-life is the time for the body to reduce the drug concentration in blood plasma by half through metabolism and excretion. It determines dosing frequency and how long effects last after stopping a medication.
How many half-lives until a substance is essentially gone?
After 7 half-lives, approximately 0.78% of the original remains. After 10 half-lives, about 0.098% remains — typically negligible for practical purposes. The exact threshold depends on the context.
Related Resources
Related Calculators
- Exponent Calculator — Solve radioactive decay equations.
- Quadratic Formula Calculator — Work with polynomial systems.
External Authority Resources
- Wolfram MathWorld: Half-Life — Mathematical decay modeling.
- Wikipedia: Half-life — Overview of physics and chemistry decay constants.
Sources & Citations
- Krane, K.S. Introductory Nuclear Physics. Wiley. 1988.
- NIST — Radioactive Half-Lives
- Libby WF. Radiocarbon Dating. University of Chicago Press. 1955.
Frequently Asked Questions
N(t) = N₀ × (½)^(t/t½), where N₀ is the initial quantity, Nₜ is the remaining amount at time t, and t½ is the half-life period. This describes exponential decay.
Carbon-14 has a half-life of approximately 5,730 years, making it the basis of radiocarbon dating for organic materials up to ~50,000 years old.
The half-life (t½) is the time for half the atoms to decay. The mean lifetime (τ) is the average time a single atom survives before decaying. They are related by: τ = t½ / ln(2) ≈ 1.4427 × t½.
The decay constant λ represents the probability per unit time that an atom decays. It relates to half-life by: λ = ln(2) / t½ ≈ 0.6931 / t½.
After 7 half-lives ~0.78% remains, after 10 half-lives ~0.098% remains. After about 10 half-lives the substance is negligible for most practical purposes.
No — half-life applies to any exponential decay including drug metabolism in the body, chemical reactions, and population decline. A drug with a 4-hour half-life has 50% concentration remaining after 4 hours.