Half-Life Calculator — Radioactive Decay and First-Order Reactions
Calculate radioactive decay, remaining quantity, or elapsed time using half-life. Works for nuclear decay, drug elimination, and any first-order decay process.
Half-life is one of the most intuitive concepts in all of science: the time it takes for half of something to disappear. It shows up in nuclear physics (radioactive decay), pharmacology (drug elimination), and chemistry (first-order reactions). The CalcHub Half-Life Calculator solves for any unknown — remaining quantity, elapsed time, or half-life itself — given the other two values.
The Key Equations
For a first-order decay process:
N(t) = N₀ × (1/2)^(t/t½)
Which is equivalent to:
N(t) = N₀ × e^(−λt), where λ = ln(2)/t½
| Variable | Meaning |
|---|---|
| N₀ | Initial quantity |
| N(t) | Remaining quantity at time t |
| t | Elapsed time |
| t½ | Half-life |
| λ | Decay constant |
How to Use the Calculator
- Open CalcHub and go to the Half-Life Calculator.
- Enter any two of: initial quantity, remaining quantity, elapsed time, and half-life.
- The tool solves for the missing variable.
- Quantity can be in grams, atoms, moles, Becquerels, or any proportional unit.
Worked Examples
How much of a 100 g sample of C-14 remains after 11,460 years? (t½ = 5,730 years)N(t) = 100 × (1/2)^(11460/5730) = 100 × (1/2)² = 100 × 0.25 = 25 g
(Exactly 2 half-lives → one quarter remains.)
A drug has a half-life of 4 hours. After how long will 90% be eliminated?Remaining = 10%, so N(t)/N₀ = 0.10
0.10 = (1/2)^(t/4)
ln(0.10) = (t/4) × ln(0.5)
t = 4 × ln(0.10)/ln(0.5) = 4 × 3.322 = 13.3 hours
Half-Lives of Common Isotopes
| Isotope | Half-Life | Application |
|---|---|---|
| Carbon-14 (¹⁴C) | 5,730 years | Archaeological dating |
| Uranium-238 (²³⁸U) | 4.47 × 10⁹ years | Geological dating |
| Iodine-131 (¹³¹I) | 8.02 days | Thyroid treatment |
| Technetium-99m (⁹⁹ᵐTc) | 6.01 hours | Medical imaging |
| Radon-222 (²²²Rn) | 3.82 days | Environmental hazard |
| Fluorine-18 (¹⁸F) | 110 minutes | PET scanning |
Multiple Half-Lives at a Glance
| Half-lives elapsed | Fraction remaining | % remaining |
|---|---|---|
| 1 | 1/2 | 50% |
| 2 | 1/4 | 25% |
| 3 | 1/8 | 12.5% |
| 5 | 1/32 | 3.1% |
| 10 | 1/1024 | 0.1% |
Why is radioactive decay always first-order?
Because the probability that any single nucleus decays in a given time interval is constant and independent of all other nuclei. This is the statistical definition of a first-order process. No matter how many atoms you have, a fixed fraction decays per unit time.
Does half-life change with temperature or pressure?
For chemical reactions, yes — half-life depends on temperature through the rate constant. For nuclear decay, no — radioactive half-lives are completely unaffected by temperature, pressure, or chemical environment. That's what makes them reliable as geological and archaeological clocks.
What is carbon dating and how does it use half-life?
Living organisms continuously exchange carbon with the atmosphere, maintaining a roughly constant ratio of ¹⁴C to ¹²C. When an organism dies, the ¹⁴C decays without replenishment. Measuring the remaining ¹⁴C ratio and knowing the half-life (5,730 years) lets you calculate time of death — accurate to about 50,000 years back.
Related calculators: Rate of Reaction Calculator · Electrochemistry Calculator · Percent Yield Calculator