To find the original mass of the Bismuth-212 sample, we can use the radioactive decay formula. The amount of substance remaining after a certain time can be calculated using the formula:
\[ N = N_0 \left(\frac{1}{2}\right)^{\frac{t}{T_{1/2}}} \]
where:
- \( N \) is the remaining mass of the sample after time \( t \),
- \( N_0 \) is the original mass of the sample,
- \( T_{1/2} \) is the half-life of the substance,
- \( t \) is the time elapsed.
Given:
- \( N = 2.64 \) grams
- \( T_{1/2} = 60.5 \) seconds
- \( t = 121 \) seconds
First, we need to determine how many half-lives have passed in the time period of 121 seconds:
\[ \text{Number of half-lives} = \frac{t}{T_{1/2}} = \frac{121 \text{ seconds}}{60.5 \text{ seconds}} \approx 2 \]
Now we can substitute this value back into the decay formula. Since approximately 2 half-lives have passed, the remaining mass is:
\[ N = N_0 \left(\frac{1}{2}\right)^2 \]
This can be rewritten as:
\[ N = N_0 \left(\frac{1}{4}\right) \]
Now we can solve for \( N_0 \):
\[ N_0 = N \times 4 \] \[ N_0 = 2.64 \text{ grams} \times 4 = 10.56 \text{ grams} \]
Thus, the original mass of the Bismuth-212 sample was 10.56 grams.
The correct response is:
10.56 g