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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

Q.13.8 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}\textrm{C} present with the stable  carbon isotope _{6}^{12}\textrm{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}\textrm{C} , and the measured activity, the age of the specimen can be approximately estimated. This is the principle of _{6}^{14}\textrm{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.

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best_answer

Since we know that activity is proportional to the number of radioactive isotopes present in the sample.

\frac{R}{R_{0}}=\frac{N}{N_{0}}=\frac{9}{15}=0.6

Also

N=N_{0}e^{-\lambda t}

\\t=-\frac{1}{\lambda }ln\frac{N}{N_{0}}\\ t=-\frac{1}{\lambda }ln0.6\\ t=\frac{0.51}{\lambda }

but \lambda = \frac{0.693}{T_{1/2}}

Therefore 

\\t=0.51\times \frac{T_{1/2}}{0.693}\\ \\t=0.735T_{1/2}

\\t\approx 4217

The age of the Indus-Valley civilisation calculated using the given specimen is approximately 4217 years.

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