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.
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