To determine the initial quantity of a radioactive substance, given its final quantity, half-life, and the elapsed time, we can use the principles of radioactive decay. Here’s a step-by-step solution to the problem:
1. Understand the given values:
- Final quantity ([tex]\( Q_f \)[/tex]) = 50 grams
- Half-life of the substance ([tex]\( t_{1/2} \)[/tex]) = 102 seconds
- Time elapsed ([tex]\( t \)[/tex]) = 300 seconds
2. Calculate the number of half-lives that have passed:
[tex]\[
\text{Number of half-lives} = \frac{\text{Time elapsed}}{\text{Half-life}}
\][/tex]
Plug in the values:
[tex]\[
\text{Number of half-lives} = \frac{300 \text{ seconds}}{102 \text{ seconds}} \approx 2.9412
\][/tex]
3. Use the radioactive decay formula to find the initial quantity:
The formula for radioactive decay is:
[tex]\[
Q_f = Q_i \left( \frac{1}{2} \right)^n
\][/tex]
where [tex]\( Q_f \)[/tex] is the final quantity, [tex]\( Q_i \)[/tex] is the initial quantity, and [tex]\( n \)[/tex] is the number of half-lives.
Rearrange this formula to solve for the initial quantity ([tex]\( Q_i \)[/tex]):
[tex]\[
Q_i = Q_f \left( 2 \right)^n
\][/tex]
Substitute the values we have:
[tex]\[
Q_i = 50 \left( 2 \right)^{2.9412}
\][/tex]
4. Calculate the initial quantity ([tex]\( Q_i \)[/tex]):
First, find [tex]\( 2^{2.9412} \)[/tex]:
[tex]\[
2^{2.9412} \approx 7.6804
\][/tex]
Now, multiply by the final quantity:
[tex]\[
Q_i = 50 \times 7.6804 = 384.02 \text{ grams}
\][/tex]
So, the correct answer is:
```
384.02 g
```
