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?

Answer 1

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.