Question

If a radioactive nuclide has a half-life of 1.3 billion years and it is determined that a rock contains 25% of its original amount of this nuclide, about how old is the rock?

1.3 billion years
2.6 billion years
0.65 billion years
3.9 billion years

Answer 1

\[\large y=Ce^{kt}\]
where \(y\) is the amount of the radioactive material at time \(t\), \(C\) is the initial amount (at \(t=0\)), \(k\) is the relative decay factor, and \(t\) is time (in billions of years).

We know the half-life is 1.3 billion years, which means after 1.3 billion years, we have half of the initial amount:
\[\large y=\frac{C}{2}=Ce^{1.3k}~~\Rightarrow~~\frac{1}{2}=e^{1.3k}\]
Solve for \(k\):
\[\large\begin{align*}
\frac{1}{2}&=e^{1.3k}\\
\ln\frac{1}{2}&=1.3k\\
k&=\frac{1}{1.3}\ln\frac{1}{2}\\
k&\approx -0.533
\end{align*}\]
We want to find out how old the rock is that contains 25% of its original amount. In other words, solve for \(t\):
\[\large 0.25=e^{kt}\]
where \(k\approx-0.533\).