Question

The half-life of plutonium-239 is 24,300 years. If a nuclear bomb released 8 kg of this
isotope, how many years would pass before the amount is reduced to 1 kg?

Answer 1

To find out how many years it will take for 8 kg of plutonium-239 to reduce to 1 kg, we can use the concept of half-lives.

The formula for the remaining amount \( N \) after a certain number of half-lives \( n \) is given by:

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

Where:
- \( N_0 \) is the initial amount (8 kg in this case)
- \( N \) is the remaining amount (1 kg in this case)

We can set up the equation:

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

Dividing both sides by 8 gives:

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

We know that:

\[
\frac{1}{8} = \left(\frac{1}{2}\right)^3
\]

Thus,

\[
\left(\frac{1}{2}\right)^n = \left(\frac{1}{2}\right)^3
\]

This implies:

\[
n = 3
\]

Now, since each half-life of plutonium-239 is 24,300 years, the total time \( t \) it will take to go through 3 half-lives is:

\[
t = n \times \text{half-life}
\]
\[
t = 3 \times 24,300 \text{ years} = 72,900 \text{ years}
\]

Hence, it will take **72,900 years** for the amount of plutonium-239 to reduce from 8 kg to 1 kg.