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.
- 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 \(n = 3\).
- 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.