NEET-XII-Physics
46: The Nucleus
- #33When charcoal is prepared from a living tree, it shows a disintegration rate of 15.3 disintegrations of 14C per gram per minute. A sample from an ancient piece of charcoal shows 14C activity to be 12.3 disintegrations per gram per minute. How old is this sample? Half-life of 14C is 5730 y.Ans : Given:
Initial activity of charcoal, A0 = 15.3 disintegrations per gram per minute
Half-life of charcoal, `` {T}_{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}``= 5730 years
Final activity of charcoal after a few years, A = 12.3 disintegrations per gram per minute
Disintegration constant, `` \lambda =\frac{0.693}{{T}_{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}`` = `` \frac{0.693}{5370}{\,\mathrm{\,y\,}}^{-1}``
Let the sample take a time of t years for the activity to reach 12.3 disintegrations per gram per minute.
Activity of the sample, `` A={A}_{0}{e}^{-\lambda t}``
`` A={A}_{0}{e}^{-\frac{0.693}{5730}\times t}``
`` \Rightarrow \,\mathrm{\,In\,}\frac{12.3}{15.3}=\frac{-0.693}{5730}t``
`` \Rightarrow 0.218253=\frac{0.693}{5730}\times t``
`` \Rightarrow t=1804.3\,\mathrm{\,years\,}``
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