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The normal activity of living carbon-containing matter is found to be about 15 decays per minute for every gram of carbon. This activity arises from the small proportion of radioactive\[{}_{6}^{14}C\]present with the stable carbon isotope \[{}_{6}^{12}C\]. When the organism is dead, its interaction with the atmosphere (which maintains the above equilibrium activity) ceases and its activity begins to drop. From the known half-life (5730 years) of\[{}_{6}^{14}C\], and the measured activity, the age of the specimen can be approximately estimated. This is the principle of \[{}_{6}^{14}C\]dating used in archaeology. Suppose a specimen from Mohenjodaro gives an activity of 9 decays per minute per gram of carbon. Estimate the approximate age of the Indus-Valley civilisation.
The normal activity of living carbon-containing matter is found to be about 15 decays per minute for every gram of carbon. This activity arises from the small proportion of radioactive\[{}_{6}^{14}C\]present with the stable carbon isotope \[{}_{6}^{12}C\]. When the organism is dead, its interaction with the atmosphere (which maintains the above equilibrium activity) ceases and its activity begins to drop. From the known half-life (5730 years) of\[{}_{6}^{14}C\], and the measured activity, the age of the specimen can be approximately estimated. This is the principle of \[{}_{6}^{14}C\]dating used in archaeology. Suppose a specimen from Mohenjodaro gives an activity of 9 decays per minute per gram of carbon. Estimate the approximate age of the Indus-Valley civilisation.
CBSE | Class 12 | NCERT | Nuclei | Physics
The normal activity of living carbon-containing matter is found to be about 15 decays per minute for every gram of carbon. This activity arises from the small proportion of radioactive\[{}_{6}^{14}C\]present with the stable carbon isotope \[{}_{6}^{12}C\]. When the organism is dead, its interaction with the atmosphere (which maintains the above equilibrium activity) ceases and its activity begins to drop. From the known half-life (5730 years) of\[{}_{6}^{14}C\], and the measured activity, the age of the specimen can be approximately estimated. This is the principle of \[{}_{6}^{14}C\]dating used in archaeology. Suppose a specimen from Mohenjodaro gives an activity of 9 decays per minute per gram of carbon. Estimate the approximate age of the Indus-Valley civilisation.

Answer –

We are given the decay rate of living carbon-containing matter, R = 15 decay/min

Let N represent the number of radioactive atoms present in a normal carbon-containing matter.

Half- life of \[{}_{6}^{14}C\]is given by \[{{T}_{\frac{1}{2}}}\] = 5730 years

We are also given the decay rate of the specimen obtained from the Mohenjo-Daro site which is

R’ = 9 decays/min

Let N’ represent the number of radioactive atoms present during the Mohenjo-Daro period in the specimen.

We know that the relation between the decay constant, λ and time, t is given by:

\[\frac{N}{{{N}_{0}}}=\frac{R}{{{R}^{‘}}}={{e}^{-\lambda t}}\]

Where, λ = Decay constant and t represents Time. So, from the above two equations we have –

\[{{e}^{-\lambda t}}=\frac{9}{15}=\frac{3}{5}\]

\[-\lambda t=\log \left( \frac{3}{5} \right)=-0.5108\]

Using the relation –

\[-\lambda =\frac{0.693}{T}\]

We then have –

\[t=0.5108\times \frac{T}{0.693}=4223.5Tyears\]

Therefore, the approximate age of the Indus-Valley civilization is 4223.5 years.

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