Question

What was the original mass of the Bismuth-212 sample if, after 121 seconds, it now weighs 2.64 grams, given that Bismuth-212 has a half-life of 60.5 seconds?

Answer 1

To find the original mass of the Bismuth-212 sample, we can use the formula for radioactive decay based on half-life. The formula is:

\[
N(t) = N_0 \left( \frac{1}{2} \right)^{\frac{t}{T_{1/2}}}
\]

where:
- \( N(t) \) is the remaining quantity of the substance at time \( t \),
- \( N_0 \) is the original quantity of the substance,
- \( T_{1/2} \) is the half-life of the substance,
- \( t \) is the elapsed time.

In this case:
- \( N(t) = 2.64 \) grams (the remaining mass after 121 seconds),
- \( T_{1/2} = 60.5 \) seconds (the half-life of Bismuth-212),
- \( t = 121 \) seconds (the total time elapsed).

First, we compute how many half-lives have elapsed during the 121 seconds. This is done by dividing the total time by the half-life:

\[
n = \frac{t}{T_{1/2}} = \frac{121 \text{ seconds}}{60.5 \text{ seconds}} \approx 2
\]

Now, using the decay formula, we set it up to solve for \( N_0 \):

\[
N(t) = N_0 \left( \frac{1}{2} \right)^n
\]

Substituting the known values:

\[
2.64 = N_0 \left( \frac{1}{2} \right)^2
\]

Calculating \( \left( \frac{1}{2} \right)^2 \):

\[
\left( \frac{1}{2} \right)^2 = \frac{1}{4}
\]

Now substituting that back into the equation:

\[
2.64 = N_0 \cdot \frac{1}{4}
\]

To solve for \( N_0 \), we can multiply both sides by 4:

\[
N_0 = 2.64 \times 4 = 10.56 \text{ grams}
\]

Thus, the original mass of the Bismuth-212 sample was:

\[
\boxed{10.56 \text{ grams}}
\]