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. $C^{12}$ is stable while its isotope $C^{14}$ is radioactive which decays to $C^{12}$. This $C^{14}$ 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 $C^{14}$ isotope. \[{}_7N^{14}+{}_0n^1 \to {}_6C^{14}+{}_1H^1\]

Atmosphere contains radioactive carbon dioxide ${}^{14}CO_2$ and normal carbon dioxide ${}^{12}CO_2$. All plants and animals take carbon from food and atmosphere. The ratio of ${}^{14}C$ and ${}^{12}C$ is fixed in living bodies. But when they die, the intake of carbon from food and atmosphere stops. Then, ${}^{12}C$ remains unchanged but ${}^{14}C$ decays to ${}^{12}C$. Hence, ${}^{14}C/{}^{12}C$ decreases with time after they die.

Let the number of atoms of ${}_6C^{14}$ in a body initially be $N_0$ when it meets its death at $t=0$. After time $t$ of its death, let the body contains the number of atoms of ${}_6C^{14}$ and ${}_6C^{12}$ equal to $N_{14}$ and $N_{12}$ 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 $\text{ln}$, \[\text{ln}e^{λt}=\text{ln}\left(\frac{N_{12}}{N_{14}}+1\right)\] \[λt=\text{ln}\left(\frac{N_{12}}{N_{14}}+1\right)\] \[t=\frac{1}{λ} \text{ln}\left(\frac{N_{12}}{N_{14}}+1\right)\] Where, \[λ=\frac{ln(2)}{T_{1/2}}\] And, $T_{1/2}$ of carbon is $5730$ years.

By knowing $N_{12}/N_{14}$, we can calculate how long ago it died i.e. the age of the specimen.

[Also See: Bohr’s Atomic Model]

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