Question

Plutonium-239, a radioactive isotope often used as fuel in nuclear reactors, has a half-life of 24,100 years. Approximately how long would it take for 504 grams of plutonium-239 to decay to just 63 grams?

Answer 1

To determine how long it will take for 504 grams of plutonium-239 to decay to 63 grams, we first need to find out how many half-lives it takes to go from 504 grams to 63 grams.

The formula for the amount of substance remaining after a certain number of half-lives is:

\[
N = N_0 \left(\frac{1}{2}\right)^n
\]

Where:
- \( N \) is the remaining amount of substance,
- \( N_0 \) is the initial amount of substance,
- \( n \) is the number of half-lives.

In this case:

- \( N_0 = 504 \) grams
- \( N = 63 \) grams

We need to find \( n \):

\[
63 = 504 \left(\frac{1}{2}\right)^n
\]

Dividing both sides by 504 gives:

\[
\frac{63}{504} = \left(\frac{1}{2}\right)^n
\]

Calculating \( \frac{63}{504} \):

\[
\frac{63}{504} = \frac{1}{8}
\]

This means:

\[
\left(\frac{1}{2}\right)^n = \frac{1}{8}
\]

Recognizing that \( \frac{1}{8} = \left(\frac{1}{2}\right)^3 \), we have:

\[
n = 3
\]

Now, since each half-life of plutonium-239 is 24,100 years, the total time (\( t \)) for 3 half-lives is:

\[
t = n \times \text{half-life} = 3 \times 24,100 \text{ years} = 72,300 \text{ years}
\]

Therefore, it would take approximately **72,300 years** for 504 grams of plutonium-239 to decay to 63 grams.