Answer :
To determine how many hours it takes for a 3000-gram sample of a radioactive isotope with a half-life of 22.5 minutes to decay down to 5 grams, we can follow these steps:
1. Understand the Concept of Half-life:
- A half-life is the time required for half of the radioactive isotope to decay. For this problem, the half-life is 22.5 minutes.
2. Calculate the Number of Half-lives Needed:
- We need to figure out how many half-lives it takes for the sample to decay from 3000 grams to 5 grams.
- The formula used to find the number of half-lives ([tex]\(n\)[/tex]) is:
[tex]\[
n = \frac{\log(\text{final mass} / \text{initial mass})}{\log(0.5)}
\][/tex]
- Plugging in the values:
[tex]\[
n = \frac{\log(5 / 3000)}{\log(0.5)}
\][/tex]
- This calculation gives us approximately 9.23 half-lives.
3. Convert the Number of Half-lives to Minutes:
- Multiply the number of half-lives by the half-life duration to find the total time in minutes:
[tex]\[
\text{Total time in minutes} = n \times 22.5
\][/tex]
- With [tex]\(n\)[/tex] being approximately 9.23, the total time in minutes is about 207.65 minutes.
4. Convert Minutes to Hours:
- Finally, convert the total time from minutes to hours by dividing by 60:
[tex]\[
\text{Total time in hours} = \frac{\text{Total time in minutes}}{60}
\][/tex]
- This results in approximately 3.46 hours.
Thus, it takes about 3.46 hours for the sample to decay from 3000 grams to 5 grams.
1. Understand the Concept of Half-life:
- A half-life is the time required for half of the radioactive isotope to decay. For this problem, the half-life is 22.5 minutes.
2. Calculate the Number of Half-lives Needed:
- We need to figure out how many half-lives it takes for the sample to decay from 3000 grams to 5 grams.
- The formula used to find the number of half-lives ([tex]\(n\)[/tex]) is:
[tex]\[
n = \frac{\log(\text{final mass} / \text{initial mass})}{\log(0.5)}
\][/tex]
- Plugging in the values:
[tex]\[
n = \frac{\log(5 / 3000)}{\log(0.5)}
\][/tex]
- This calculation gives us approximately 9.23 half-lives.
3. Convert the Number of Half-lives to Minutes:
- Multiply the number of half-lives by the half-life duration to find the total time in minutes:
[tex]\[
\text{Total time in minutes} = n \times 22.5
\][/tex]
- With [tex]\(n\)[/tex] being approximately 9.23, the total time in minutes is about 207.65 minutes.
4. Convert Minutes to Hours:
- Finally, convert the total time from minutes to hours by dividing by 60:
[tex]\[
\text{Total time in hours} = \frac{\text{Total time in minutes}}{60}
\][/tex]
- This results in approximately 3.46 hours.
Thus, it takes about 3.46 hours for the sample to decay from 3000 grams to 5 grams.
