State the law of radioactive decay. Derive N = N₀ exp(−λt) and the relation between half-life and decay constant. Give two uses of radioactivity.

1. 1 · State the law. The number of atoms decaying per second is proportional to the number present: dN/dt = −λN.
2. 2 · Integrate. dN/N = −λ dt gives ln(N/N₀) = −λt, so N = N₀ exp(−λt).
3. 3 · Define half-life. At t = T½, N = N₀/2.
4. 4 · Relate T½ and λ. ½ = exp(−λT½), so λT½ = ln 2 = 0.693 and T½ = 0.693/λ.
5. 5 · Draw the decay curve. [Graph: N against t; starts at N₀ and falls to N₀/2, N₀/4, N₀/8 at T½, 2T½, 3T½, never touching the time axis]
6. 6 · Give two uses. Carbon dating of fossils; cobalt-60 gamma rays for cancer treatment.
7. 7 · Close in one line. So, each half-life halves what is left, and the sample never vanishes completely.

Scoring tip: Show the integration in two lines; the examiner looks for ln(N/N₀) = −λt. Mark N₀/2 and T½ on the curve.