#### Question

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 `""_6^14C` present with the stable carbon isotope `""_6^12C` . 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 `""_6^14C` and the measured activity, the age of the specimen can be approximately estimated. This is the principle of `""_6^14C"` 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.

#### Solution

Decay rate of living carbon-containing matter, *R* = 15 decay/min

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

Half life of `""_6^14C` `T_(1/2)`= 5730 years

The decay rate of the specimen obtained from the Mohenjodaro site:

*R*' = 9 decays/min

Let N' be the number of radioactive atoms present in the specimen during the Mohenjodaro period.

Therefore, we can relate the decay constant, *λ*and time, *t* as:

`N/N' = R/R' = e^(-lambdat)`

`e^(-lambdat) = 9/15 = 3/5`

`-lambdat = log_e 3/5 = -0.5108`

`:. t= 0.5108/lambda`

But `lambda = 0.639/T_"1/2" = 0.693/5730`

`:, t = 0.5108/(0.693/5730) = 4223.5 year`

Hence, the approximate age of the Indus-Valley civilisation is 4223.5 years.