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?(1 point)
Responses
24,100 years
24,100 years
72,300 years
72,300 years
96,400 years
96,400 years
48,200 years
Responses
24,100 years
24,100 years
72,300 years
72,300 years
96,400 years
96,400 years
48,200 years
Answers
GPT-4o mini
To determine how long it takes for 504 grams of plutonium-239 to decay to 63 grams, we can use the concept of half-lives.
1. Calculate how many half-lives it takes to go from 504 grams to 63 grams.
Starting amount = 504 grams
Final amount = 63 grams
To find the number of half-lives (n) that have occurred, we can use the formula:
\[
\text{Final amount} = \text{Initial amount} \times \left(\frac{1}{2}\right)^n
\]
Plugging in the known values:
\[
63 = 504 \times \left(\frac{1}{2}\right)^n
\]
Dividing both sides by 504:
\[
\frac{63}{504} = \left(\frac{1}{2}\right)^n
\]
Simplifying the fraction:
\[
\frac{63}{504} = \frac{1}{8}
\]
Now we need to express \(\frac{1}{8}\) as a power of \(\frac{1}{2}\):
\[
\frac{1}{8} = \left(\frac{1}{2}\right)^3
\]
Thus, we can see that \(n = 3\).
2. Now, we need to find the total time for these 3 half-lives:
Since the half-life of plutonium-239 is 24,100 years, the total time for 3 half-lives is:
\[
3 \times 24,100 \text{ years} = 72,300 \text{ years}
\]
Thus, it would take **72,300 years** for 504 grams of plutonium-239 to decay to 63 grams.
The answer is **72,300 years**.
1. Calculate how many half-lives it takes to go from 504 grams to 63 grams.
Starting amount = 504 grams
Final amount = 63 grams
To find the number of half-lives (n) that have occurred, we can use the formula:
\[
\text{Final amount} = \text{Initial amount} \times \left(\frac{1}{2}\right)^n
\]
Plugging in the known values:
\[
63 = 504 \times \left(\frac{1}{2}\right)^n
\]
Dividing both sides by 504:
\[
\frac{63}{504} = \left(\frac{1}{2}\right)^n
\]
Simplifying the fraction:
\[
\frac{63}{504} = \frac{1}{8}
\]
Now we need to express \(\frac{1}{8}\) as a power of \(\frac{1}{2}\):
\[
\frac{1}{8} = \left(\frac{1}{2}\right)^3
\]
Thus, we can see that \(n = 3\).
2. Now, we need to find the total time for these 3 half-lives:
Since the half-life of plutonium-239 is 24,100 years, the total time for 3 half-lives is:
\[
3 \times 24,100 \text{ years} = 72,300 \text{ years}
\]
Thus, it would take **72,300 years** for 504 grams of plutonium-239 to decay to 63 grams.
The answer is **72,300 years**.