#### Question

When charcoal is prepared from a living tree, it shows a disintegration rate of 15.3 disintegrations of ^{14}C per gram per minute. A sample from an ancient piece of charcoal shows ^{14}C activity to be 12.3 disintegrations per gram per minute. How old is this sample? Half-life of ^{14}C is 5730 y.

#### Solution

Given:

Initial activity of charcoal, A_{0} = 15.3 disintegrations per gram per minute

Half-life of charcoal, `T_"1/2"` = 5730 years

Final activity of charcoal after a few years, A = 12.3 disintegrations per gram per minute

Disintegration constant, `lambda = 0.693/T_"1/2" = 0.693/5370 "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_0e^(-lambdat)`

`A = A_0e^(-0.693/5730 xx t)`

⇒`"In" 12.3/15.3 = (-0.693)/5730 t`

⇒ `0.218253 = 0.693/5730 t`

⇒ t = 1804.3 years