Question

In an unweathered sample of igneous rock, the ratio of an unstable isotope to its stable daughter isotope is 1:15. If no daughters were present at the time the rock cooled below closure temperature, and the half-life of the isotope is 50 million years, how old is the rock?

Answer 1

200 million years

Step-by-step explanation:

The equation that describes the decay of a radioactive isotope is

`N(t)=N_0 ((1)/(2))^{(t)/(t_(1/2))}`

where

`N(t)` is the amount of radioactive isotope left at time t

`N_0` is the initial amount of isotope

`t_(1/2)` is the half-life of the sample

In this problem, the ratio between unstable isotope and daughter isotope is 1:15; this means that

`(N(t))/(N_0)=(1)/(16)`

Because the "total proportion" of original sample was 1+15=16.

Also we know that the half-life is

`t_(1/2)=50\cdot 10^6 y`

So we can re-arrange the equation to find t, the age of the rock:

`t=t_(1/2) log_(0.5)((N)/(N_0))=(50\cdot 10^6)log_(0.5)((1)/(16))=200\cdot 10^6 y`

So, 200 million years.