A piece of wood form the ruins of an ancient building was found to have a `C^(14)` activity of 12 disintegrations per minute per gram of its carbon content. The `C^(14)` activity of the living wood is 16 disintegrations/minute/gram. How long ago did the trees, from which the wooden sample came, die? Given half-life of `C^(14)` is 5760 years.

Options

A. 1,672 years
B. 2,391 years
C. 3,291 years
D. 4,453 years

Answer

B

Explanation

`tau_(1//2) = 5,760` years
`t=n tau_(1//2) rArr n=5,760 t`
Let `N_(0)` and N be the activities of `C^(14)` at t=0 and time t, respectively
`N=12, N_(0)=16`
Note that activity is proportional to number of nuclei.
`N/N_(0)=(1/2)^(n)`
`12/16=1/2^(n) rArr 2^(n)=4/3`
n ln(2)=ln (4/3)
`t/5,760 ln(2) =2ln(2) -ln(3)`
`t=5,760 (2ln(2)-ln(3))/(ln(2))`years
=2,391 years