To determine the original mass of the Bismuth-212 sample, we can use the formula for exponential decay based on the half-life:
\[ N(t) = N_0 \left( \frac{1}{2} \right)^{\frac{t}{t_{1/2}}} \]
Where:
- \(N(t)\) is the remaining quantity (2.64 g),
- \(N_0\) is the original quantity we want to find,
- \(t\) is the elapsed time (121 seconds),
- \(t_{1/2}\) is the half-life (60.5 seconds).
Now we rearrange the formula to solve for \(N_0\):
\[ N_0 = N(t) \left( \frac{1}{2} \right)^{-\frac{t}{t_{1/2}}} \]
Substituting the values into the equation:
\[ N_0 = 2.64 \left( \frac{1}{2} \right)^{-\frac{121}{60.5}} \]
Calculating the exponent:
\[ \frac{121}{60.5} \approx 2 \]
Then, the equation becomes:
\[ N_0 = 2.64 \left( \frac{1}{2} \right)^{-2} \]
Since \(\left( \frac{1}{2} \right)^{-2} = 4\), we have:
\[ N_0 = 2.64 \times 4 = 10.56 \text{ g} \]
Therefore, the original mass of the Bismuth-212 sample was 10.56 g.
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