Ncert Exercises & Solutions

A piece of wood from the ruins of an ancient building was found to have a ¹⁴C activity of 12 disintegrations per minute per gram of its carbon content. The ¹⁴C activity of the living wood is 16 disintegrations per minute per gram. How long ago did the tree, from which the wooden sample came, die? Given half-life of ¹⁴C is 5760 yr.

Hint: The decay constant depends on the half-life of the element.

Step 1 : Use the decay law .

Given, R = 12 dis / \min per g, R_{0} = 16 dis / \min per g, T_{1 / 2} = 5760 yr

Let t be the span of the tree .

According to radioactive decay law,

R = R_{0} e^{- λt} or \frac{R}{R_{0}} = e^{- λt} or e^{λt} = \frac{R_{0}}{R}

Step 2 : Find t .

Taking \log on the both the sides :

λt \log_{e} e = \log_{e} \frac{R_{0}}{R} \Rightarrow λt = (\log_{10} \frac{16}{12}) \times 2.303

t = \frac{2.303 (\log 4 - \log 3)}{λ}

= \frac{2.303 (0.6020 - 4.771) \times 5760}{0.6931} (∵ λ = \frac{0.6931}{T_{1 / 2}})

= 2391.20 yr

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