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Calculate remaining amount, initial amount, half-life, or elapsed time
Same units as time
Half-life is the time required for exactly half of a substance to decay or be eliminated. After one half-life, 50% remains; after two half-lives, 25% remains; after three, 12.5% remains, and so on.
The formula A = A₀ × (1/2)^(t/h) calculates the remaining amount (A) after time t, given initial amount A₀ and half-life h. The exponent (t/h) represents how many half-lives have passed.
Carbon-14 has a half-life of 5,730 years (carbon dating). Uranium-238: 4.5 billion years. Medical isotopes like Technetium-99m: 6 hours. Caffeine in the body: 5 hours. Drug medications also have characteristic half-lives.
Mathematically, the substance never completely disappears. However, after 10 half-lives, less than 0.1% remains (1/1024). In practice, this is often considered negligible. Medical professionals often use 5 half-lives (3% remaining) as "effectively eliminated."
Yes! If you know the initial amount, remaining amount, and time elapsed, you can calculate the half-life using: h = t × ln(2) / ln(A₀/A). This is useful for experimental determination of half-lives.
The half-life and time must use the same units (both years, both hours, etc.). The amount can be in any units (grams, atoms, becquerels, etc.) as long as initial and remaining amounts use the same units.
The decay constant (k) and half-life (h) are related by: h = ln(2) / k ≈ 0.693 / k. The decay constant is used in the continuous decay formula A = A₀e^(-kt), while half-life is used in A = A₀(1/2)^(t/h).
For radioactive decay and first-order processes, the half-life is constant regardless of how much substance you have. Whether you start with 1000 grams or 10 grams, half will decay in the same amount of time. This makes half-life a fundamental property of each isotope.