• Tags: carbon-dating, age, uranium
• Difficulty: Moderate

Radioactive dating uses the known half-life to determine age. Carbon-14 dating (t_$\frac{1}{2}$ = 5730 years): living organisms maintain a constant C-14/C-12 ratio through exchange with the atmosphere. After death, C-14 decays: N = $N_0$e^(-lambdat). Measuring the current C-14 activity and comparing with the initial activity gives the age: t = $\frac{1}{lambda}$*ln$\frac{A_0}{A}$ = $\frac{t_(1/2}{0.693}$)*ln$\frac{A_0}{A}$. Effective range: up to ~50,000 years. For geological dating $\frac{millions}{billions of years}$, use longer-lived isotopes: U-238 (t_$\frac{1}{2}$ = 4.5 x 10^9 years) decays to Pb-206, K-40 (t_$\frac{1}{2}$ = 1.28 x 10^9 years) decays to Ar-40. In uranium-lead dating: if $N_{Pb}$ atoms of Pb-206 are found alongside $N_U$ atoms of U-238, then $N_0$ = $N_U$ + $N_{Pb}$, and t = $\frac{1}{lambda}$*ln(1 + $N_{Pb}$/$N_U$). JEE typically gives the ratio and asks for age in terms of half-life.