Answer :
To find the amount of Astatine-218 left from a 500 gram sample after 10 seconds, we can use the concept of exponential decay, where the quantity decreases over time at a rate proportional to its current value.
Here's a step-by-step solution:
1. Identify Key Information:
- The initial amount of Astatine-218 is 500 grams.
- The half-life of Astatine-218 is 2 seconds.
- We want to find the remaining amount after 10 seconds.
2. Understand Half-Life:
- Half-life is the time it takes for half of a sample to decay. For Astatine-218, this time is 2 seconds.
3. Define the Exponential Decay Formula:
The formula for exponential decay is:
[tex]\[
N(t) = N_0 \times e^{-kt}
\][/tex]
where:
- [tex]\( N(t) \)[/tex] is the amount remaining after time [tex]\( t \)[/tex].
- [tex]\( N_0 \)[/tex] is the initial amount.
- [tex]\( k \)[/tex] is the decay constant.
- [tex]\( e \)[/tex] is the base of the natural logarithm (approximately 2.71828).
- [tex]\( t \)[/tex] is the time elapsed.
4. Calculate the Decay Constant ([tex]\( k \)[/tex]):
The decay constant [tex]\( k \)[/tex] can be determined from the half-life formula:
[tex]\[
k = \frac{\ln(2)}{\text{half-life}}
\][/tex]
Substituting the half-life of 2 seconds:
[tex]\[
k = \frac{\ln(2)}{2}
\][/tex]
5. Find Remaining Quantity After 10 Seconds:
Using the exponential decay formula:
- Substitute [tex]\( N_0 = 500 \)[/tex] grams, [tex]\( t = 10 \)[/tex] seconds, and the calculated [tex]\( k \)[/tex].
6. Calculate:
- Compute and round the result to three decimal places.
After performing the calculations, the remaining amount of Astatine-218 after 10 seconds is found to be approximately 15.625 grams. This means that from the initial 500 grams, only about 15.625 grams remain after 10 seconds due to the radioactive decay.
Here's a step-by-step solution:
1. Identify Key Information:
- The initial amount of Astatine-218 is 500 grams.
- The half-life of Astatine-218 is 2 seconds.
- We want to find the remaining amount after 10 seconds.
2. Understand Half-Life:
- Half-life is the time it takes for half of a sample to decay. For Astatine-218, this time is 2 seconds.
3. Define the Exponential Decay Formula:
The formula for exponential decay is:
[tex]\[
N(t) = N_0 \times e^{-kt}
\][/tex]
where:
- [tex]\( N(t) \)[/tex] is the amount remaining after time [tex]\( t \)[/tex].
- [tex]\( N_0 \)[/tex] is the initial amount.
- [tex]\( k \)[/tex] is the decay constant.
- [tex]\( e \)[/tex] is the base of the natural logarithm (approximately 2.71828).
- [tex]\( t \)[/tex] is the time elapsed.
4. Calculate the Decay Constant ([tex]\( k \)[/tex]):
The decay constant [tex]\( k \)[/tex] can be determined from the half-life formula:
[tex]\[
k = \frac{\ln(2)}{\text{half-life}}
\][/tex]
Substituting the half-life of 2 seconds:
[tex]\[
k = \frac{\ln(2)}{2}
\][/tex]
5. Find Remaining Quantity After 10 Seconds:
Using the exponential decay formula:
- Substitute [tex]\( N_0 = 500 \)[/tex] grams, [tex]\( t = 10 \)[/tex] seconds, and the calculated [tex]\( k \)[/tex].
6. Calculate:
- Compute and round the result to three decimal places.
After performing the calculations, the remaining amount of Astatine-218 after 10 seconds is found to be approximately 15.625 grams. This means that from the initial 500 grams, only about 15.625 grams remain after 10 seconds due to the radioactive decay.
