Question
After 121 seconds, how many grams of the sample will remain if Bismuth-212 has a half-life of 60.5 seconds?
Answers
GPT-4o mini
To determine the remaining amount of Bismuth-212 after 121 seconds, we can use the half-life formula. The half-life of a radioactive isotope is the time it takes for half of the sample to decay.
1. **Calculate the number of half-lives**:
We find the total time elapsed divided by the half-life:
\[
\text{Number of half-lives} = \frac{\text{Total time}}{\text{Half-life}} = \frac{121 \text{ seconds}}{60.5 \text{ seconds}} \approx 2
\]
This means that approximately 2 half-lives have passed.
2. **Determine the remaining fraction of the sample**:
After each half-life, the amount of the substance remaining is halved. Therefore, after \( n \) half-lives, the remaining amount can be expressed as:
\[
\text{Remaining amount} = \text{Initial amount} \times \left(\frac{1}{2}\right)^n
\]
For \( n = 2 \):
\[
\text{Remaining amount} = \text{Initial amount} \times \left(\frac{1}{2}\right)^2 = \text{Initial amount} \times \frac{1}{4}
\]
3. **Final calculation**:
If we assume an initial amount of Bismuth-212 (let's denote it as \( m_0 \)), the remaining amount after 121 seconds will be:
\[
\text{Remaining amount} = m_0 \times \frac{1}{4}
\]
Therefore, after 121 seconds, only a quarter of the original sample remains.
If you have the initial mass of the sample, you can substitute that value into \( m_0 \) to find out how many grams remain. If \( m_0 \) was, for example, 100 grams, then the remaining mass would be:
\[
\text{Remaining amount} = 100 \text{ grams} \times \frac{1}{4} = 25 \text{ grams}
\]
In conclusion, after 121 seconds, \( \frac{1}{4} \) of the initial amount of Bismuth-212 will remain.
1. **Calculate the number of half-lives**:
We find the total time elapsed divided by the half-life:
\[
\text{Number of half-lives} = \frac{\text{Total time}}{\text{Half-life}} = \frac{121 \text{ seconds}}{60.5 \text{ seconds}} \approx 2
\]
This means that approximately 2 half-lives have passed.
2. **Determine the remaining fraction of the sample**:
After each half-life, the amount of the substance remaining is halved. Therefore, after \( n \) half-lives, the remaining amount can be expressed as:
\[
\text{Remaining amount} = \text{Initial amount} \times \left(\frac{1}{2}\right)^n
\]
For \( n = 2 \):
\[
\text{Remaining amount} = \text{Initial amount} \times \left(\frac{1}{2}\right)^2 = \text{Initial amount} \times \frac{1}{4}
\]
3. **Final calculation**:
If we assume an initial amount of Bismuth-212 (let's denote it as \( m_0 \)), the remaining amount after 121 seconds will be:
\[
\text{Remaining amount} = m_0 \times \frac{1}{4}
\]
Therefore, after 121 seconds, only a quarter of the original sample remains.
If you have the initial mass of the sample, you can substitute that value into \( m_0 \) to find out how many grams remain. If \( m_0 \) was, for example, 100 grams, then the remaining mass would be:
\[
\text{Remaining amount} = 100 \text{ grams} \times \frac{1}{4} = 25 \text{ grams}
\]
In conclusion, after 121 seconds, \( \frac{1}{4} \) of the initial amount of Bismuth-212 will remain.