To find the initial mass of Fermium-253 given its half-life and the remaining mass after a specified time, we can use the concept of radioactive decay. The key formula involved here is the exponential decay model, which can be described as:
[tex]\[ \text{remaining mass} = \text{initial mass} \times \left( \frac{1}{2} \right)^{\frac{t}{T}} \][/tex]
where:
- [tex]\(\text{remaining mass} = 0.50 \, \text{grams}\)[/tex] (the amount of Fermium-253 remaining after time [tex]\( t \)[/tex])
- [tex]\(\text{initial mass}\)[/tex] is what we need to find
- [tex]\( t = 1.336 \, \text{seconds}\)[/tex] (the elapsed time)
- [tex]\( T = 0.334 \, \text{seconds}\)[/tex] (the half-life of Fermium-253)
Step-by-Step Solution:
1. Identify the number of half-lives ([tex]\( \frac{t}{T} \)[/tex]) passed during the elapsed time [tex]\( t \)[/tex]:
[tex]\[
\frac{t}{T} = \frac{1.336 \text{ seconds}}{0.334 \text{ seconds}} = 4
\][/tex]
This means that 4 half-lives have passed in 1.336 seconds.
2. Plug the number of half-lives ([tex]\( n \)[/tex]) into the exponential decay formula:
[tex]\[
\text{remaining mass} = \text{initial mass} \times \left( \frac{1}{2} \right)^n
\][/tex]
where [tex]\( n = 4 \)[/tex].
3. Re-organize the formula to solve for the initial mass:
[tex]\[
\text{initial mass} = \text{remaining mass} \div \left( \frac{1}{2} \right)^n
\][/tex]
Substitute the known values:
[tex]\[
\text{initial mass} = 0.50 \text{ grams} \div \left( \frac{1}{2} \right)^4
\][/tex]
4. Calculate [tex]\( \left( \frac{1}{2} \right)^4 \)[/tex]:
[tex]\[
\left( \frac{1}{2} \right)^4 = \frac{1}{16}
\][/tex]
5. Determine the initial mass:
[tex]\[
\text{initial mass} = 0.50 \text{ grams} \div \frac{1}{16} = 0.50 \text{ grams} \times 16 = 8 \text{ grams}
\][/tex]
Therefore, the initial mass of Fermium-253 was [tex]\( 8 \, \text{grams} \)[/tex].
