Question

Analysis of a rock sample shows that it contains 6.25% of its original uranium-235. How old is the rock? How do you know?

Answer 1

Answer:

$2.82\cdot 10^9 y$

Explanation:

A radioactive isotope is an isotope that undergoes nuclear decay, breaking apart into a smaller nucleus and emitting radiation during the process.

The half-life of an isotope is the amount of time it takes for a certain quantity of a radioactive isotope to halve.

For a radioactive isotope, the amount of substance left after a certain time t is:

$m(t)=m_0 (\frac{1}{2})^{\frac{t}{\tau}}$ (1)

where

$m_0$ is the mass of the substance at time t = 0

m(t) is the mass of the substance at time t

$\tau$ is the half-life of the isotope

In this problem, the isotope is uranium-235, which has a half-life of

$\tau=7.04\cdot 10^8 y$

We also know that the amount of uranium left in the rock sample is 6.25% of its original value, this means that

$\frac{m(t)}{m_0}=\frac{6.25}{100}$

Substituting into (1) and solving for t, we can find how much time has passed:

$t=-\tau log_2 (\frac{m(t)}{m_0})=-(7.04\cdot 10^8) log_2 (\frac{6.25}{100})=2.82\cdot 10^9 y$