Half-Life Decay

Also known as radioactive decay · carbon dating

N=N0(12)t/t1/2N = N_0 \left(\frac{1}{2}\right)^{t/t_{1/2}}

Worked example: C-14: 100 units after 11460 yr (2 half-lives) → 25 remain — press Try an example to run it live, then adjust anything.

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Grade 11Grade 11 Math — Functions & Applications

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Half-Life Decay explained

N0t1/2tN

Radioactive decay never runs out of sample all at once — it halves, and halves again, forever. Each half-life leaves exactly 50% of what entered it, so after two half-lives a quarter remains, after ten about a thousandth. Carbon-14, with its 5,730-year half-life, is the famous example: a bone whose C-14 content has fallen to one quarter of the living value has been dead roughly 11,460 years. N and N0N_0 can be in any matching unit — grams, atom counts, or becquerels — since only their ratio matters.

Solving for time or half-life inverts the exponential with a base-2 logarithm, which demands 0 < N < N0N_0: you cannot take the log of a zero ratio, and decay never leaves more than it started with. The same mathematics governs drug elimination in pharmacology and the six-hour half-life of technetium-99m, the workhorse isotope of medical imaging.

Half-Life Decay formula

N=N0(12)t/t1/2N = N_0 \left(\frac{1}{2}\right)^{t/t_{1/2}}
Where
  • NN= Remaining quantity
  • N0N_0= Initial quantity
  • tt= Elapsed time (s)
  • t1/2t_{1/2}= Half-life (s)

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