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

Answer:

200 million years

Explanation:

The equation that describes the decay of a radioactive isotope is

$N(t)=N_0 (\frac{1}{2})^{\frac{t}{t_{1/2}}}$

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

$\frac{N(t)}{N_0}=\frac{1}{16}$

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:

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

So, 200 million years.