Half Life Calculator
Calculate radioactive decay parameters including half-life, remaining amount, time elapsed, and decay constant. Works for radioactive isotopes, carbon dating, and first-order chemical reactions.
What is half-life?
Half-life (t½) is the time required for half of a radioactive substance to decay. After one half-life, 50% remains; after two half-lives, 25% remains; after three, 12.5% remains, and so on. Each radioactive isotope has a characteristic half-life ranging from fractions of a second to billions of years. Half-life is independent of the initial amount.
How do you calculate remaining amount after time t?
Use N(t) = N₀ × (1/2)^(t/t½), where N₀ is initial amount, t is time elapsed, and t½ is half-life. Alternatively, use N(t) = N₀ × e^(-λt), where λ = ln(2)/t½ = 0.693/t½ is the decay constant. Both formulas give the same result; choose based on what information you have.
What is the decay constant (λ)?
The decay constant (λ) is the probability per unit time that a nucleus will decay. It relates to half-life by: λ = ln(2)/t½ = 0.693147/t½. The decay law is N(t) = N₀e^(-λt). A larger λ means faster decay and shorter half-life. Units of λ are inverse time (s⁻¹, year⁻¹, etc.).
How many half-lives until a substance is "gone"?
Theoretically, radioactive decay never reaches exactly zero. After 10 half-lives, only 0.098% (about 1/1000) remains. After 20 half-lives, only 0.0001% remains. In practice, 10 half-lives is often considered sufficient for the substance to be negligible. The number of half-lives is n = t/t½ or n = ln(N₀/N)/ln(2).
What are some common half-lives?
Carbon-14: 5,730 years (used for dating), Uranium-238: 4.5 billion years, Iodine-131: 8 days (medical use), Technetium-99m: 6 hours (medical imaging), Radon-222: 3.8 days, Plutonium-239: 24,110 years, Tritium (H-3): 12.3 years. These vast differences make different isotopes useful for different applications.
How is half-life used in carbon dating?
Living organisms maintain constant C-14/C-12 ratio by exchanging carbon with the environment. When they die, C-14 decays (t½ = 5,730 years) while C-12 remains constant. By measuring the remaining C-14 fraction, we calculate time since death: t = -t½ × ln(N/N₀)/ln(2). Accurate up to ~50,000 years (about 9 half-lives).
Does half-life change with temperature or pressure?
No! Nuclear decay half-life is completely independent of external conditions like temperature, pressure, chemical state, or physical form. This is because decay is a nuclear process, not a chemical one. The nucleus is unaffected by electrons or molecular environment. This constancy makes radioactive decay ideal for dating and timekeeping.
What is the difference between half-life and mean lifetime?
Half-life (t½) is the time for half to decay. Mean lifetime (τ) is the average time a nucleus exists before decaying: τ = 1/λ = t½/ln(2) ≈ 1.443 × t½. After one mean lifetime, 1/e ≈ 36.8% remains (not 50%). Both describe decay rate but are used in different contexts; half-life is more intuitive.
How do you calculate half-life from decay data?
Measure initial amount (N₀) and amount after time t (N). Use t½ = t × ln(2) / ln(N₀/N). Example: If 1000 atoms decay to 250 atoms in 20 days, t½ = 20 × 0.693 / ln(1000/250) = 20 × 0.693 / ln(4) = 20 × 0.693 / 1.386 = 10 days.
Can half-life be used for chemical reactions?
Yes! First-order chemical reactions follow the same mathematics as radioactive decay. For a first-order reaction with rate constant k, the half-life is t½ = ln(2)/k = 0.693/k. This applies to drug metabolism, radioactive decay, and any process where the rate depends on concentration. Unlike nuclear decay, chemical t½ CAN vary with temperature (via k).