Physics

Physics

Problem

A radioactive sample initially contains 8.0×10208.0\times10^{20} nuclei. Its decay constant is 0.693 day−10.693\,\text{day}^{-1}. Calculate the approximate number of undecayed nuclei left after 33 days. a) 2.0×10202.0\times10^{20} nuclei b) 1.0×10201.0\times10^{20} nuclei c) 0.50×10200.50\times10^{20} nuclei d) 4.0×10204.0\times10^{20} nuclei
Intuitive explanation

This is about exponential decay — the key insight is that radioactive nuclei disappear at a rate proportional to how many remain, which gives you the exponential formula N(t)=N0e−λtN(t) = N_0 e^{-\lambda t}. The decay constant λ\lambda tells you how fast the process happens, and you just plug in your time to find what fraction survives. Notice that λ=0.693\lambda = 0.693 is suspiciously close to ln⁡(2)\ln(2), which hints at a useful shortcut.

Intuitive explanation

This is about exponential decay — the key insight is that radioactive nuclei disappear at a rate proportional to how many remain, which gives you the exponential formula N(t)=N0e−λtN(t) = N_0 e^{-\lambda t}. The decay constant λ\lambda tells you how fast the process happens, and you just plug in your time to find what fraction survives. Notice that λ=0.693\lambda = 0.693 is suspiciously close to ln⁡(2)\ln(2), which hints at a useful shortcut.

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