Answer :
To solve this problem, we need to determine how long it takes for a 3000 gram sample of a radioactive isotope to decay down to 5 grams, given that the half-life of the isotope is 22.5 minutes. We'll break down the process step by step:
1. Understand the Concept of Half-Life:
The half-life of a radioactive substance is the time it takes for half of a sample to decay. For this isotope, every 22.5 minutes, the amount of the substance remaining will be half of what it was at the start of that period.
2. Determine the Number of Half-Lives Needed:
To find out how many half-lives are required to reduce the substance from 3000 grams to 5 grams, we use the formula:
[tex]\[
\text{Number of half-lives} = \frac{\log(\text{final\_amount / initial\_amount})}{\log(0.5)}
\][/tex]
By substituting the values, we find the number of half-lives is approximately 9.23.
3. Calculate the Total Time in Minutes:
Since each half-life is 22.5 minutes, we calculate the total time in minutes by multiplying the number of half-lives by the half-life duration:
[tex]\[
\text{Total time in minutes} = \text{Number of half-lives} \times \text{Half-life in minutes}
\][/tex]
This gives us a total time of approximately 207.65 minutes.
4. Convert the Total Time from Minutes to Hours:
To convert the time from minutes to hours, divide the total minutes by 60:
[tex]\[
\text{Total time in hours} = \frac{\text{Total time in minutes}}{60}
\][/tex]
This results in approximately 3.46 hours.
Therefore, it takes approximately 3.46 hours for a 3000 gram sample of the radioactive isotope to decay to 5 grams.
1. Understand the Concept of Half-Life:
The half-life of a radioactive substance is the time it takes for half of a sample to decay. For this isotope, every 22.5 minutes, the amount of the substance remaining will be half of what it was at the start of that period.
2. Determine the Number of Half-Lives Needed:
To find out how many half-lives are required to reduce the substance from 3000 grams to 5 grams, we use the formula:
[tex]\[
\text{Number of half-lives} = \frac{\log(\text{final\_amount / initial\_amount})}{\log(0.5)}
\][/tex]
By substituting the values, we find the number of half-lives is approximately 9.23.
3. Calculate the Total Time in Minutes:
Since each half-life is 22.5 minutes, we calculate the total time in minutes by multiplying the number of half-lives by the half-life duration:
[tex]\[
\text{Total time in minutes} = \text{Number of half-lives} \times \text{Half-life in minutes}
\][/tex]
This gives us a total time of approximately 207.65 minutes.
4. Convert the Total Time from Minutes to Hours:
To convert the time from minutes to hours, divide the total minutes by 60:
[tex]\[
\text{Total time in hours} = \frac{\text{Total time in minutes}}{60}
\][/tex]
This results in approximately 3.46 hours.
Therefore, it takes approximately 3.46 hours for a 3000 gram sample of the radioactive isotope to decay to 5 grams.
