The half-life of plutonium-238 is 87.7 years.

Question

What percentage of the atoms in a sample of plutonium-238 will remain radioactive after 263.1 years?
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What percentage of the atoms in the same sample of plutonium-238 will have changed to another isotope after 263.1 years?

Answer 1

To determine the percentage of plutonium-238 that remains radioactive after a certain period, we use the concept of half-lives. The formula to find the remaining quantity of a radioactive substance after a certain period is:

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

where:

• $N$ is the remaining quantity of the substance,
• $N_0$ is the initial quantity of the substance,
• $t$ is the elapsed time,
• $t_{1/2}$ is the half-life of the substance.

In this case, the half-life ($t_{1/2}$) of plutonium-238 is 87.7 years, and the elapsed time ($t$) is 263.1 years. First, we calculate the number of half-lives that have passed:

$$\text{Number of half-lives}=\frac{t}{t_{1/2}}=\frac{263.1}{87.7}\approx 3$$

Now, we can calculate the remaining percentage of the original sample:

$$N=N_0{\left(\frac{1}{2}\right)}^3=N_0\left(\frac{1}{8}\right)$$

This means that after 263.1 years, $\frac{1}{8}$ of the original plutonium-238 remains:

$$\text{Percentage remaining}=\left(\frac{1}{8}\right)\times 100$$

Therefore, after 263.1 years, approximately 12.5% of the plutonium-238 atoms will remain radioactive.

To find the percentage of atoms that have changed to another isotope, we subtract the remaining percentage from 100%:

$$\text{Percentage changed}=100$$

Thus, after 263.1 years, approximately 87.5% of the atoms in the sample of plutonium-238 will have changed to another isotope.