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.

#### 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 `""_6^14"C"`, `"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"^(-lambda"t")`

`"e"^(-lambda"t") = 9/15 = 3/5`

`-lambda"t" = 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 years

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