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

Radioactive dating uses the known half-life to determine age. Carbon-14 dating (t_(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_0e^(-lambdat). Measuring the current C-14 activity and comparing with the initial activity gives the age: t = (1/lambda)*ln(A_0/A) = (t_(1/2)/0.693)*ln(A_0/A). Effective range: up to ~50,000 years. For geological dating (millions/billions of years), use longer-lived isotopes: U-238 (t_(1/2) = 4.5 x 10^9 years) decays to Pb-206, K-40 (t_(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 = (1/lambda)*ln(1 + N_Pb/N_U). JEE typically gives the ratio and asks for age in terms of half-life.