Half-Life Calculator

Frequently Asked Questions

What is the half-life formula?

N = N₀ · (1/2)^(t / t½). Equivalently N = N₀·e^(−λt) with decay constant λ = ln2 / t½. The tool solves for any one unknown.

What is the decay constant and mean lifetime?

λ = ln2 / t½ is the per-unit-time decay probability; the mean lifetime τ = 1/λ = t½ / ln2 ≈ 1.4426 · t½.

How many half-lives until something is "gone"?

After n half-lives, (1/2)ⁿ remains: ~50%, 25%, 12.5%… After 10 half-lives under 0.1% remains. It never reaches exactly zero mathematically.

Does this apply to medication?

The same math models drug elimination (plasma half-life), but dosing is clinical - this is educational, not medical advice.

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How This Calculator Works

Exponential decay follows N = N₀ · (½)^t÷t½, where N₀ is the starting quantity, t½ is the half-life (the time for the amount to fall by half), and t is elapsed time. The calculator solves for any one of the four unknowns by algebraically inverting that relation: the remaining amount is a direct power; elapsed time and half-life come from the logarithmic form t = t½ · ln(N₀÷N) ÷ ln2; the initial amount divides back through the decay factor. It also reports two derived constants every physicist uses: the decay constant λ = ln2 ÷ t½ (the instantaneous fractional decay rate) and the mean lifetime τ = 1÷λ = t½ ÷ ln2 ≈ 1.4427 × t½, the average time an individual atom survives.

Worked Example: Carbon-14 Dating

Carbon-14 has a half-life of 5730 years. A sample retains 25% of its original ¹⁴C. How old is it? Using t = t½ · ln(N₀÷N) ÷ ln2 with N₀÷N = 100÷25 = 4: t = 5730 × ln(4) ÷ 0.6931 = 5730 × 1.3863 ÷ 0.6931 = 5730 × 2 = 11,460 years. That makes sense: 25% means exactly two half-lives have passed (100% → 50% → 25%). The decay constant is λ = 0.6931 ÷ 5730 ≈ 1.21 × 10⁻⁴ per year, and the mean lifetime is 5730 ÷ 0.6931 ≈ 8267 years.

Where Half-Life Is Used

Radiometric dating: ¹⁴C for organic remains up to ~50,000 years; uranium-lead and potassium-argon for rocks billions of years old.

Nuclear medicine: tracer isotopes like ⁹⁹ᵐTc (6 h) are chosen so they decay away soon after imaging, limiting patient dose.

Pharmacokinetics: drug elimination is roughly first-order; after ~5 half-lives about 97% has cleared, which sets dosing intervals.

Safety & waste: spent-fuel storage timelines hinge on the longest-lived isotopes' half-lives.

Common Mistakes & Edge Cases

Assuming decay is linear - it is exponential. Two half-lives leave 25%, not 0%; ten leave ~0.1%, never exactly zero.

Mixing time units: half-life in years but elapsed time in days. Keep both in the same unit; the ratio t÷t½ must be dimensionless.

Confusing half-life with mean lifetime. τ is about 44% longer than t½; they are not interchangeable.

Believing biological drug half-life is purely chemical - it depends on metabolism, kidney÷liver function, and other drugs; this is a generic model, not medical advice.

Requesting elapsed time when remaining ≥ initial - decay cannot increase the amount, so the log is undefined and the input is rejected.

Related Concepts & Limits

This is a single, constant-rate first-order process. Real systems can deviate: branching decay (one isotope to several daughters), in-growth of radioactive daughters (secular equilibrium), or multi-compartment drug kinetics where a single half-life is only an approximation. The math is identical to RC-circuit discharge, cooling (Newton's law), and continuously compounded depreciation - all share N = N₀e^−λt with λ = ln2÷t½. Activity (decays per second, the becquerel) equals λN, so a short half-life means high specific activity. Results show six significant figures; for ages beyond ~10 half-lives the measured fraction becomes tiny and real-world counting uncertainty dominates the precision shown here.

Frequently Asked Questions

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