Half-Life Calculator
Calculate radioactive decay using half-life. Solve for remaining amount, initial amount, half-life, or elapsed time. Includes decay constant, percent remaining, and common isotope presets.
The starting quantity before decay (activity, mass, or number of atoms). Must be greater than zero.
The time required for half of the substance to decay. Must be greater than zero.
How much time has passed since the initial measurement.
Enter the required values above to calculate decay
Table of Contents
What is Half-Life?
Half-life (t½) is the time required for half of a radioactive substance to decay. It is a fundamental concept in nuclear physics, chemistry, and medicine. Key points:
- After one half-life, 50% of the original substance remains
- After two half-lives, 25% remains; after three, 12.5% remains
- Half-life is constant for a given isotope and does not depend on the initial amount
- Different isotopes have vastly different half-lives—from fractions of a second to billions of years
Radioactive decay follows exponential kinetics. The amount remaining decreases by the same fraction in each equal time interval, which is why half-life is a convenient way to describe the rate of decay.
Half-life is used in carbon-14 dating, medical imaging (e.g. technetium-99m), radiation safety, nuclear waste management, and pharmacokinetics (drug elimination from the body).
N = N₀ × (½)^(t/t½)
How it Works
Choose what to solve for, then enter the other three known values:
- Remaining amount: Enter initial amount, half-life, and elapsed time
- Initial amount: Enter remaining amount, half-life, and elapsed time
- Half-life: Enter initial amount, remaining amount, and elapsed time
- Elapsed time: Enter initial amount, remaining amount, and half-life
The calculator automatically computes:
- Fraction and percent of substance remaining
- Number of half-lives elapsed (n = t / t½)
- Decay constant λ = ln(2) / t½
- All results using the exponential decay equation
Time and half-life use the same unit (seconds, minutes, hours, days, or years). First-order decay is assumed; the initial amount can represent activity, mass, or particle count.
Formulas
Remaining Amount
N = N₀ × (½)^(t/t½)
Where N₀ is the initial amount, t is elapsed time, and t½ is the half-life.
Number of Half-Lives
n = t / t½
The number of half-lives elapsed is the ratio of elapsed time to half-life.
Elapsed Time
t = t½ × log₂(N₀/N)
Solve for time when initial amount, remaining amount, and half-life are known.
Half-Life from Measurements
t½ = t × ln(2) / ln(N₀/N)
When you know how much decayed over a known time interval.
Decay Constant
λ = ln(2) / t½
The decay constant λ relates to half-life and appears in N = N₀e^(-λt).
Examples
Example 1: Carbon-14 after 11,460 years
Given: N₀ = 1000 g, t½ = 5,730 years, t = 11,460 years (2 half-lives)
- n = 11,460 / 5,730 = 2 half-lives
- N = 1000 × (½)² = 1000 × 0.25 = 250 g
- 25% of the original sample remains
Example 2: Technetium-99m in nuclear medicine
Given: N₀ = 500 MBq, t½ = 6 hours, t = 12 hours
- N = 500 × (½)² = 125 MBq
- After 12 hours (two half-lives), activity drops to 25% of the initial dose
Example 3: Find elapsed time
Given: N₀ = 800, N = 100, t½ = 10 days
- t = 10 × log₂(800/100) = 10 × log₂(8) = 10 × 3 = 30 days
- Three half-lives have passed
Common Use Cases
- Radiocarbon Dating: Estimate age of organic materials using carbon-14 half-life (5,730 years)
- Nuclear Medicine: Plan imaging and therapy doses based on isotope half-lives (e.g. Tc-99m, I-131)
- Radiation Safety: Predict how long until activity drops to a safe level
- Pharmacokinetics: Model drug elimination when clearance follows first-order kinetics
- Nuclear Waste: Assess long-term decay of spent fuel and fission products
- Education: Teach exponential decay and logarithmic relationships in science courses
- Research: Calculate sample activity after storage or transport delays
Frequently Asked Questions
The remaining amount is N = N₀ × (1/2)^(t/t½), where N₀ is initial amount, t is elapsed time, and t½ is half-life. Equivalently, N = N₀ × e^(-λt) with decay constant λ = ln(2)/t½.
For first-order (exponential) decay, the probability that any given atom decays in the next instant is fixed. The rate is proportional to the amount present, so the fraction lost per unit time is constant—hence a fixed half-life for each isotope.
Yes, when drug elimination follows first-order kinetics, the plasma concentration halves over each elimination half-life. Many medications are described this way in pharmacology.
In theory decay never reaches exactly zero. In practice, after about 10 half-lives less than 0.1% remains—often considered effectively gone for radiation and dating purposes.
Use any consistent unit for both half-life and elapsed time. This calculator lets you pick seconds, minutes, hours, days, or years for both values.
No decay has occurred (t = 0), or you need different inputs. To solve for half-life or elapsed time, remaining must be less than initial.