CALTRX

Half-Life Calculator

Calculate radioactive or exponential decay using half-life. See the complete solution with step-by-step working and formula explanations.

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Used in Time/Half-Life modes

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Tips & Notes

  • ✓After n half-lives, the remaining quantity = N₀ / 2ⁿ. After 3 half-lives = N₀/8 = 12.5% remains.
  • ✓Half-lives are constant regardless of initial amount — a property unique to first-order exponential decay.
  • ✓ln(2) ≈ 0.693 appears in all half-life formulas: T½ = 0.693/λ and λ = 0.693/T½.
  • ✓Carbon-14 dating uses T½=5,730 years. After 57,300 years (10 half-lives), only 0.1% remains.
  • ✓Drug half-life: after 5 half-lives, ~97% of a drug is eliminated from the body (1/2⁵ = 3.125% remains).

Common Mistakes

  • ✗Using wrong time units — T½ and t must be in the same units (both days, or both years, etc.).
  • ✗Confusing half-life with mean lifetime. Mean lifetime τ = T½/ln(2) ≈ 1.443 × T½.
  • ✗Applying half-life formula to non-exponential decay — half-life is constant only for first-order reactions.
  • ✗Computing 100 × 0.5 = 50 for each step instead of multiplying cumulatively. After 2 half-lives: 100 → 50 → 25, not 0.
  • ✗Forgetting that half-life applies to the remaining amount, not the total initial amount.

Half-Life Calculator Overview

The half-life is the time required for exactly half of a substance to decay, transform, or be eliminated. It is the defining characteristic of radioactive isotopes, drug pharmacokinetics, and any first-order exponential decay process. The concept is powerful because the half-life is constant regardless of the starting amount — whether you begin with 1 gram or 1 kilogram, exactly half remains after one half-life, a quarter after two, an eighth after three. This property makes half-life the most reliable and intuitive way to characterize exponential decay.

The fundamental half-life formula:

N(t) = N₀ × (1/2)^(t/T½)
Where N₀ is the initial quantity, t is elapsed time, and T½ is the half-life period.
EX: 100g substance with T½=5 days. After 15 days (3 half-lives): N = 100×(1/2)³ = 100/8 = 12.5g
Equivalent exponential form using the decay constant λ:
N(t) = N₀ × e^(−λt), where λ = ln(2)/T½ ≈ 0.693/T½
EX: Iodine-131, T½=8 days → λ=0.693/8=0.0866 per day. After 24 days: N = N₀×e^(−0.0866×24) = N₀×0.125 = 12.5% remains
Finding elapsed time from remaining quantity:
t = T½ × log₂(N₀/N) = T½ × ln(N₀/N) / ln(2)
EX: 80g decays to 10g with T½=3 hours → t = 3×log₂(80/10) = 3×log₂(8) = 3×3 = 9 hours
After n half-lives: N = N₀ / 2ⁿ. After 10 half-lives: N = N₀/1024 ≈ 0.1% remains. The substance is practically gone after 7 half-lives (<1%). This rule of thumb guides safe handling of radioactive materials and drug clearance timelines. Drug half-life: after 5 half-lives, approximately 97% of a drug is eliminated (1/2⁵ = 3.125% remains). Caffeine with T½≈5 hours: 200mg at 2 PM → ~6.25mg by midnight. Dosing intervals are typically 1-2 half-lives to maintain therapeutic levels without accumulation. The constancy of half-life is a direct consequence of quantum mechanics. Radioactive decay is a fundamentally probabilistic process — each nucleus has a fixed probability per unit time of decaying, independent of how long it has already existed or what is happening to neighboring nuclei.

Frequently Asked Questions

Half-life is the time required for exactly half of a radioactive substance to decay. After one half-life: 50% remains. After two: 25%. After three: 12.5%. After n half-lives: (1/2)ⁿ of the original remains. Formula: N(t) = N₀ × (1/2)^(t/t₁/₂). Example: Iodine-131 has a half-life of 8.02 days. Starting with 100 mg: after 24 days (3 half-lives) = 100 × (1/2)³ = 12.5 mg remaining.

Half-life is related to the decay constant λ by: t₁/₂ = ln(2)/λ = 0.6931/λ. The decay equation is N(t) = N₀ × e^(−λt). These are equivalent: (1/2)^(t/t₁/₂) = e^(−λt). Example: Carbon-14 has λ = 1.21 × 10⁻⁴ per year. t₁/₂ = 0.6931/(1.21×10⁻⁴) = 5730 years. A sample with 1000 atoms after 5730 years: N = 1000 × e^(−1.21×10⁻⁴ × 5730) = 1000 × 0.5 = 500 atoms.

Radiocarbon dating measures the ratio of Carbon-14 to Carbon-12. Living organisms continuously absorb C-14 from the atmosphere; when they die, C-14 decays with a half-life of 5730 years without replenishment. Measuring the remaining C-14 ratio gives the time since death. Formula: t = (t₁/₂ / ln2) × ln(N₀/N). If a sample has 25% of its original C-14: t = 5730/0.6931 × ln(1/0.25) = 8267 × 1.386 = 11,460 years old. The method is accurate for samples up to about 50,000 years old.

Half-life (t½) is the time after which exactly 50% of the original amount remains. Mean lifetime (τ, tau) is the average time an individual atom or molecule exists before decay — it is the time at which approximately 36.8% (1/e) of the sample remains. The relationship is: mean lifetime = half-life / ln(2) ≈ half-life × 1.4427. For carbon-14 with half-life 5730 years, the mean lifetime is about 8267 years. Half-life is more intuitive for practical calculations; mean lifetime appears in physics equations because it simplifies exponential decay formulas from N = N₀(1/2)^(t/t½) to N = N₀e^(-t/τ).

Yes — half-life applies to any first-order process where the rate of decrease is proportional to the current amount. In pharmacokinetics, drug half-life determines how often a medication must be taken: a drug with a 6-hour half-life reaches steady state after about 5 half-lives (30 hours) and clears to less than 3 percent after 5 half-lives. In finance, the concept appears in depreciation and the decay of market information. In biology, mRNA half-lives range from minutes to hours and determine how quickly cells can respond to signals. Carbon-14 dating uses a 5730-year half-life to date organic materials up to about 50,000 years old. The mathematics is identical across all these applications.

Rearrange the half-life formula. Starting from N = N0 times (1/2) to the power of (t divided by t_half), solve for N0: N0 = N divided by (1/2) to the power of (t divided by t_half), which equals N times 2 to the power of (t divided by t_half). Example: a sample now contains 25 grams of a substance with a 10-year half-life. The sample is 30 years old. N0 = 25 times 2 to the power of (30 divided by 10) = 25 times 2 cubed = 25 times 8 = 200 grams initially. This works for any first-order decay — just substitute the appropriate half-life and use the same formula.