Half-Life Calculator

Calculate remaining quantity, half-life, or elapsed time using N = N₀·(½)^(t/t½).

⚛️ Nuclear📐 N=N₀(½)^(t/t½)☢️ Radioactive Decay
Initial quantity (N₀)
Half-life (t½)
Elapsed time (t)
⚠️ Enter valid positive numbers (N must be less than N₀).

What Is Half-Life?

The half-life (t½) is the time for exactly half of a radioactive sample to decay. After one half-life: 50% remains. After two: 25%. After n half-lives: N = N₀ × (½)ⁿ = N₀ × (½)^(t/t½). This exponential decay applies to radioactive nuclei, drug concentrations in the body, and any first-order process.

Half-lives span enormous ranges: polonium-214: 164 μs; radon-222: 3.8 days; carbon-14: 5,730 years; uranium-238: 4.47 billion years (comparable to Earth's age). The decay constant λ = ln(2)/t½ = 0.693/t½ connects the half-life to the more fundamental exponential form N = N₀e^(−λt).

Radiocarbon dating uses C-14 (t½ = 5,730 years) to date organic materials up to ~50,000 years old. Living organisms maintain a constant C-14/C-12 ratio (from atmospheric CO₂). After death, C-14 decays. Measuring the remaining ratio gives the time since death: t = (t½/ln2) × ln(N₀/N).

Activity (A) is the decay rate: A = λN = (ln2/t½) × N, measured in Becquerels (Bq; 1 Bq = 1 decay/s) or Curies (Ci; 1 Ci = 3.7×10¹⁰ Bq). Both N (quantity) and A (activity) halve with each half-life. A 1 GBq sample becomes 0.5 GBq after one half-life.

Formula Reference Table

Solve ForFormulaNotes
Remaining quantityN = N₀ × (½)^(t/t½)Same units as N₀
Exponential formN = N₀ · e^(−λt)λ = ln(2)/t½ = 0.693/t½
Number of half-livesn = t/t½N = N₀/2ⁿ
Time elapsedt = t½ · log₂(N₀/N)= t½·ln(N₀/N)/ln2
Half-life from λt½ = ln(2)/λ = 0.693/λλ = decay constant (s⁻¹)
ActivityA = λN = N·ln(2)/t½Bq (decays/s) or Ci

3 Worked Examples

Example 1
Carbon-14 Dating

A bone fragment has C-14 activity 25% of a living sample. t½(C-14) = 5,730 yr.

  • N/N₀ = 0.25 = (½)^(t/5730)
  • t/5730 = log₂(1/0.25) = log₂(4) = 2
  • t = 2 × 5,730 = 11,460 years old
✓ Bone is ~11,460 years old (2 half-lives)
Example 2
Medical Isotope — Tc-99m

Tc-99m: t½ = 6.02 hr. Patient receives 500 MBq. Activity after 24 hr?

  • n = 24/6.02 = 3.99 ≈ 4 half-lives
  • A = 500 × (½)⁴ = 500/16 = 31.25 MBq
  • After 48 hr: 500/256 = 1.95 MBq (safe to discharge patient)
✓ Activity after 24 hr = 31.3 MBq (4 half-lives)
Example 3
Nuclear Waste — Find Time

Strontium-90 (t½ = 28.8 yr). How long until 1,000 g decays to 1 g?

  • N/N₀ = 1/1000 = 0.001
  • n = log₂(1000) = 9.97 half-lives
  • t = 9.97 × 28.8 = 287 years
✓ 287 years for Sr-90 to decay to 0.1% of original

Real-World Applications

🏥
Nuclear Medicine
Tc-99m (t½=6.02h), I-131 (t½=8d), F-18 (t½=110min) are used for imaging and therapy. Short half-lives minimize patient radiation dose; the isotope decays away quickly after the scan or treatment.
🌍
Geological Dating
U-238 (t½=4.47 Gy) and Th-232 (t½=14 Gy) date rocks to billions of years. Zircon crystals (retaining uranium but rejecting lead) are dated to 4.4 billion years — some of Earth's oldest material.
🔬
Archaeological Dating
C-14 dating (t½=5,730 yr) is accurate from a few hundred to ~50,000 years. Beyond that, too little C-14 remains to measure. Used to date mummies, ancient wood, seeds, charcoal — anything once-living.
⚡
Nuclear Power
Spent nuclear fuel contains fission products with various half-lives: Cs-137 (30 yr), Sr-90 (28.8 yr) are long-lived concerns; short-lived isotopes decay within weeks to years. Dry cask storage is designed for 40–100 year management.
💊
Pharmacokinetics
Drug half-life determines dosing schedules. Aspirin: t½≈15-20 min. Caffeine: t½≈5-6 hr. Diazepam: t½=20-100 hr. After 5 half-lives, >97% of a drug is eliminated. Dosing intervals are typically one half-life to maintain therapeutic levels.

Common Mistakes to Avoid

⚠️
Confusing half-life with mean lifetime

Mean lifetime τ = t½/ln(2) = 1.443 × t½. After one mean lifetime, 1/e ≈ 36.8% remains (not 50%). After one half-life, exactly 50% remains.

⚠️
Using wrong base

N = N₀(½)^(t/t½) uses base ½. The equivalent N = N₀e^(−λt) uses natural log and λ = ln2/t½. Don't mix: N₀e^(−t/t½) gives wrong result (uses wrong base).

⚠️
Time units must match

t and t½ must be in the same units. If t½ = 5,730 years and t = 11,460 years, n = 2. If one is in years and the other in days, convert first.

⚠️
Confusing activity with quantity

Both N (atoms/grams) and activity A = λN halve every half-life. But A depends on λ too: a short-lived isotope with large λ has high activity even with few atoms.

⚠️
Thinking half-life means complete decay in 2×t½

In 2 half-lives: 25% remains, not 0%. In 10 half-lives: (½)¹⁰ = 0.098% remains. True zero is never reached mathematically — only approached asymptotically.

Frequently Asked Questions

What is half-life? ▾
The time for exactly half of a radioactive sample to decay. It is constant for a given isotope, independent of sample size, temperature, or chemical state. After n half-lives: fraction remaining = (½)ⁿ. After 10 half-lives, only 1/1,024 ≈ 0.1% remains.
Why is decay exponential? ▾
Each nucleus decays independently with constant probability per unit time (λ). The more nuclei, the more decays per second (A = λN). This gives dN/dt = −λN → N = N₀e^(−λt) — the exponential decay law. The same mathematics applies to anything with constant per-unit probability of 'dying' (capacitor discharge, drug elimination).
What is the decay constant λ? ▾
λ = ln(2)/t½ is the probability per unit time that a given nucleus will decay. High λ (short t½) = fast decay. λ for U-238 = 4.9×10⁻¹⁸ s⁻¹; for Tc-99m = 3.2×10⁻⁵ s⁻¹. Activity A = λN; 1 Bq = 1 s⁻¹.
How does radiocarbon dating work? ▾
Living organisms incorporate C-14 (produced by cosmic rays + N-14 in atmosphere) at a constant rate, maintaining atmospheric C-14/C-12 ratio. After death, C-14 decays (t½=5,730 yr) while stable C-12 stays constant. Measuring the remaining ratio gives time since death. Calibration curves correct for historic atmospheric C-14 variations.
What is secular equilibrium? ▾
In a decay chain (U-238 → Th-234 → Pa-234 → ... → Pb-206), if the parent has a much longer t½ than daughters, the activity of each daughter equals the parent activity — secular equilibrium. Used in radium-226/radon-222 measurements and calibration of activity meters.
How is half-life related to nuclear stability? ▾
Short t½ = unstable nucleus, high decay energy. Long t½ = stable or semi-stable. The valley of stability (binding energy chart) shows stable nuclei. Those above or below are unstable — too many or too few neutrons. The semi-empirical mass formula predicts stability from proton/neutron count.
Can half-life be changed? ▾
Essentially no — the strong and weak nuclear forces that govern decay are unaffected by temperature, pressure, or chemical environment (with tiny exceptions: electron capture rate changes slightly with chemical bonding, affecting t½ by <0.1%). This makes radioactive dating reliable.
What is the Curie unit? ▾
1 Ci = 3.7×10¹⁰ Bq (decays/s) — originally defined as the activity of 1 gram of Ra-226. Modern unit is the Becquerel (Bq = 1 decay/s). Medical isotopes are measured in MBq (megabecquerels) or GBq. The effective dose (Sieverts) combines activity with radiation type and tissue sensitivity.

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