Radioactive Dating

Radioactive dating or carbon dating is a method which is used to estimate the age of archaeological specimens through radioactive process.

All plants and animals, living and dead contain carbon. C12 is stable while its isotope C14 is radioactive which decays to C12. This C14 isotope is formed by cosmic rays which are assumed to have hit the earth at a relatively constant rate. These cosmic rays react with nitrogen present in atmosphere and form this C14 isotope. \[{}_7N^{14}+{}_0n^1 \to {}_6C^{14}+{}_1H^1\]

Atmosphere contains radioactive carbon dioxide 14CO2 and normal carbon dioxide 12CO2. All plants and animals take carbon from food and atmosphere. The ratio of 14C and 12C is fixed in living bodies. But when they die, the intake of carbon from food and atmosphere stops. Then, 12C remains unchanged but 14C decays to 12C. Hence, 14C/12C decreases with time after they die.

Let the number of atoms of 6C14 in a body initially be N0 when it meets its death at t=0. After time t of its death, let the body contains the number of atoms of 6C14 and 6C12 equal to N14 and N12 respectively. Then, \[N_0=N_{12}+N_{14} \text{ __(1)}\] According to radioactive decay, \[N_{14}=N_0e^{-λt} \text{ __(2)}\] From equations (1) and (2), \[N_{14}=\left( N_{12}+N_{14} \right)e^{-λt} \] \[e^{λt}=\frac{N_{12}}{N_{14}}+1\] Taking ln, \[lne^{λt}=ln\left(\frac{N_{12}}{N_{14}}+1\right)\] \[λt=ln\left(\frac{N_{12}}{N_{14}}+1\right)\] \[t=\frac{1}{λ} ln\left(\frac{N_{12}}{N_{14}}+1\right)\] Where, \[λ=\frac{ln(2)}{T_{1/2}}\] And, T1/2 of carbon is 5730 years.

By knowing N12/N14, we can calculate how long ago it died i.e. the age of the specimen.

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