Half-Life Calculator

This half life calculator doubles as a radioactive decay calculator, a nuclear half life calculator, and a drug half life calculator. Pick the mode that matches what you already know, plug the numbers in, and read the answer. The math runs in your browser using the standard exponential decay formula, so there is no page reload and no rounding until the very end.

1. Pick a mode."Find Remaining Amount" answers "how much is left?". "Find Half-Life" lets you back out the half-life from a before-and-after measurement, which is how you would calculate half life from lab data. "Find Time" tells you how long it takes to decay from one amount to another.
2. Enter the initial amount (N₀). This can be grams, milligrams, atoms, counts per minute, millicuries, or any unit of quantity. The calculator does not care about the unit, only the ratio.
3. Enter the half-life. Use the time unit that fits your problem. Carbon-14 dating uses years, iodine-131 uses days, caffeine uses hours. Just keep it consistent with the elapsed time field.
4. Enter elapsed time or remaining amount. The third input depends on the mode. The result panel updates live as you type.
5. Check the reference table. The isotope table at the bottom of the widget lists common half-lives for carbon-14, uranium-238, iodine-131, radium-226, cobalt-60, tritium, plutonium-239, and radon-222. Use those numbers directly if you do not have your own.

Units must match across fields. If half-life is in years, elapsed time must be in years. If the remaining amount is larger than the initial amount, no decay has happened and the calculator will flag the inputs as invalid. For drug dosing questions, remember that the clinical half life is usually the biological half life, which already accounts for metabolism and excretion rather than pure physical decay.

The half life formula comes from the fact that decay is exponential: the rate at which atoms fall apart (or a drug is cleared) is proportional to how much is left. That gives two equivalent equations, one written with a base of 1/2 and one written with the natural exponential e. Any radioactive decay calculator or drug half life calculator uses one of these two forms.

Standard Form (Base 1/2)

N(t) = N₀ × (1/2)^(t / T)

Where:
  N(t) = amount remaining at time t
  N₀   = initial amount
  t    = elapsed time
  T    = half-life (same time units as t)

This is the form most people learn first. Every time t equals one half-life, the exponent goes up by 1 and the amount is cut in half. After 3 half-lives, N = N₀ × (1/2)³ = N₀ × 0.125, so 12.5% remains.

Exponential Form with Decay Constant λ

N(t) = N₀ × e^(−λt)

Decay constant:  λ = ln(2) / T ≈ 0.6931 / T

Where:
  e    = 2.71828...
  λ    = decay constant (per unit time)
  T    = half-life (same time units as 1/λ)

The two forms are mathematically identical. Physicists and chemists prefer the λ version because it plugs directly into differential equations and activity calculations (activity A = λ × N). If a problem gives you a decay constant of 0.121 per year, the half-life is ln(2) ÷ 0.121 ≈ 5.73 years, which is carbon-14.

Solving for Remaining Quantity, Elapsed Time, or Initial Amount

Remaining amount (given N₀, T, t):
  N = N₀ × (1/2)^(t/T)

Elapsed time (given N₀, N, T):
  t = T × log₂(N₀ / N) = T × ln(N₀ / N) / ln(2)

Half-life (given N₀, N, t):
  T = t × ln(2) / ln(N₀ / N)

Initial amount (given N, T, t):
  N₀ = N × 2^(t/T)

Worked Example 1: Caffeine After 10 Hours

Caffeine has a biological half-life of about 5 hours in a healthy adult. Drink a 100 mg coffee at 8 AM. How much is in your bloodstream at 6 PM, 10 hours later?

N₀ = 100 mg
T  = 5 hours
t  = 10 hours

N = 100 × (1/2)^(10/5)
  = 100 × (1/2)²
  = 100 × 0.25
  = 25 mg

After 10 hours (2 half-lives): 25 mg remains.
After 15 hours (3 half-lives): 12.5 mg.
After 25 hours (5 half-lives): 3.125 mg (about 3% left).

Worked Example 2: Carbon-14 Dating

A bone fragment has 25% of the carbon-14 a living animal would contain. Carbon-14's half-life is 5,730 years. How old is the bone?

N / N₀ = 0.25
T = 5,730 years

t = T × log₂(N₀ / N)
  = 5,730 × log₂(1 / 0.25)
  = 5,730 × log₂(4)
  = 5,730 × 2
  = 11,460 years

Quick Reference: Half-Lives of Common Substances

These are the values most people look up. Biological half-lives for drugs are averages for healthy adults and vary with liver function, age, and genetics.

For a back-of-the-envelope check, after n half-lives the remaining fraction is (1/2)ⁿ: 1 half-life leaves 50%, 2 leaves 25%, 3 leaves 12.5%, 4 leaves 6.25%, 5 leaves 3.125%, and 10 leaves about 0.098%.

Half-life is not just a physics curiosity. It drives prescription dosing schedules, sets the limits on radiocarbon dating, and determines how long nuclear waste must be isolated. Here is how the same equation shows up in very different real-world problems, plus the traps that catch people using a decay calculator for the first time.

Pharmacokinetics: Why Dosing Intervals Match Half-Life

A drug's biological half-life controls how often you take it. The rule of thumb is that it takes about 5 half-lives to reach steady state (when the amount going in each dose equals the amount being cleared) and another 5 to wash out after you stop. That is why SSRIs like fluoxetine (half-life 1 to 4 days, active metabolite up to 16 days) linger for weeks, while short-acting sleep aids like zaleplon (half-life 1 hour) are gone by morning.

If a drug is dosed less often than its half-life, the level drops a lot between doses. If it is dosed much more often, levels stack up until clearance catches up. Dosing every half-life keeps the peak-to-trough ratio around 2:1, which is the target range for most drugs.

Radiocarbon Dating: The 50,000-Year Ceiling

Living things constantly exchange carbon with the atmosphere, so they hold a steady ratio of carbon-14 to carbon-12. When they die, the exchange stops and the carbon-14 decays with a 5,730-year half-life. Measuring the remaining ratio in a sample gives the time since death. After about 10 half-lives (roughly 57,000 years), carbon-14 drops below what labs can distinguish from background noise. This is why radiocarbon dating tops out around 50,000 years. For older objects, dating uses potassium-argon (half-life 1.25 × 10⁹ years) or uranium-lead (4.5 × 10⁹ years), which stay measurable for billions of years.

Nuclear Waste Storage: The 10-Half-Life Rule

A common storage guideline is that radioactive material is effectively gone after 10 half-lives, when about 0.1% of the original activity is left. For iodine-131 (8 days), that is 80 days, which is why hospital patients treated with I-131 are released after a short isolation period. For cesium-137 (30 years), 10 half-lives is 300 years, which is why Chernobyl and Fukushima contamination is still an active issue decades later. For plutonium-239 (24,100 years), 10 half-lives is roughly a quarter of a million years, which is why deep geological repositories are required rather than surface storage.

Common Traps When Using a Half-Life Calculator

• One half-life means 50%, not 0%. The substance is half-gone, not gone. This is by far the most common misread.
• Two half-lives means 25%, not 0%. Each half-life halves the remaining amount, so 2 half-lives leaves (1/2)² = 25%, not (1/2 + 1/2) = 0%. Half-lives compound multiplicatively, not additively.
• Physical half-life is not biological half-life. Iodine-131 has a physical half-life of 8 days, but its biological half-life in the thyroid is about 80 days. The effective half-life combines both: 1 / T_eff = 1 / T_phys + 1 / T_bio, which gives roughly 7.3 days for I-131 in the thyroid.
• Half-life is independent of how much you start with. 1 gram and 1 kilogram of the same isotope both lose half their mass in one half-life. The absolute amount lost differs; the fraction does not.
• Units must match. If half-life is in days, elapsed time must be in days. Mixing hours and days is the second most common mistake after the 50% trap.