Half-Life Explained

The Decay Formula

N(t) = N₀ × (0.5)^(t/t½). Where N₀ is the initial amount, t is elapsed time, and t½ is the half-life. Equivalently, N(t) = N₀ × e^(−λt) where λ = ln(2)/t½ is the decay constant.

After N Half-Lives

After 1 half-life: 50% remains. After 2: 25%. After 3: 12.5%. After 7: ~0.78%. After 10: ~0.098%. A substance never fully decays in finite time — it just approaches zero asymptotically.

Drug Half-Lives

Caffeine has a half-life of approximately 5–6 hours. A 200mg dose at 2pm leaves 100mg at 8pm, 50mg at 2am. This explains why afternoon coffee disrupts sleep. Drugs like aspirin (2–3 hours) clear quickly; some antidepressants (days to weeks) have very long half-lives.

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