Question
Plutonium-239 is a radioactive isotope frequently employed as nuclear reactor fuel, with a half-life of 24,100 years. Approximately how much time would be required for 504 grams of plutonium-239 to undergo decay, leaving only 63 grams of plutonium-239 behind?
Answers
GPT-4o mini
To determine the time required for 504 grams of plutonium-239 to decay to 63 grams, we can use the concept of half-lives.
1. **Determine the number of half-lives** required to go from 504 grams to 63 grams:
We start with the initial amount, \( N_0 = 504 \) grams, and we want to find the final amount, \( N = 63 \) grams. The relationship between the initial amount, the final amount, and the number of half-lives \( n \) is given by:
\[
N = N_0 \left( \frac{1}{2} \right)^n
\]
Substituting the known values:
\[
63 = 504 \left( \frac{1}{2} \right)^n
\]
Now, divide both sides by 504:
\[
\frac{63}{504} = \left( \frac{1}{2} \right)^n
\]
Calculating \( \frac{63}{504} \):
\[
\frac{63}{504} = \frac{1}{8}
\]
Now, we can rewrite this as:
\[
\frac{1}{8} = \left( \frac{1}{2} \right)^n
\]
Since \( \frac{1}{8} = \left( \frac{1}{2} \right)^3 \), we can equate the exponents:
\[
n = 3
\]
2. **Calculate the time** for 3 half-lives:
The half-life of plutonium-239 is given as 24,100 years. Therefore, the total time \( t \) required for 3 half-lives is:
\[
t = n \times \text{half-life} = 3 \times 24,100 \text{ years}
\]
Calculating this:
\[
t = 3 \times 24,100 = 72,300 \text{ years}
\]
Thus, the time required for 504 grams of plutonium-239 to decay to 63 grams is **72,300 years**.
1. **Determine the number of half-lives** required to go from 504 grams to 63 grams:
We start with the initial amount, \( N_0 = 504 \) grams, and we want to find the final amount, \( N = 63 \) grams. The relationship between the initial amount, the final amount, and the number of half-lives \( n \) is given by:
\[
N = N_0 \left( \frac{1}{2} \right)^n
\]
Substituting the known values:
\[
63 = 504 \left( \frac{1}{2} \right)^n
\]
Now, divide both sides by 504:
\[
\frac{63}{504} = \left( \frac{1}{2} \right)^n
\]
Calculating \( \frac{63}{504} \):
\[
\frac{63}{504} = \frac{1}{8}
\]
Now, we can rewrite this as:
\[
\frac{1}{8} = \left( \frac{1}{2} \right)^n
\]
Since \( \frac{1}{8} = \left( \frac{1}{2} \right)^3 \), we can equate the exponents:
\[
n = 3
\]
2. **Calculate the time** for 3 half-lives:
The half-life of plutonium-239 is given as 24,100 years. Therefore, the total time \( t \) required for 3 half-lives is:
\[
t = n \times \text{half-life} = 3 \times 24,100 \text{ years}
\]
Calculating this:
\[
t = 3 \times 24,100 = 72,300 \text{ years}
\]
Thus, the time required for 504 grams of plutonium-239 to decay to 63 grams is **72,300 years**.