You have 54,544 grams of a radioactive kind of iodine. How much will be left after 106 minutes if its half-life is 53 minutes?
Understand the Problem
The question asks how much of a radioactive isotope remains after a certain time, given its initial mass and half-life. We need to calculate how many half-lives occur within the given time frame and then reduce the initial mass accordingly.
Answer
$13636$
Answer for screen readers
$13636$
Steps to Solve
- Calculate the number of half-lives Divide the total time by the half-life to find out how many half-lives have passed.
$$ \text{Number of half-lives} = \frac{\text{Total time}}{\text{Half-life}} $$
- Plug in the values The total time is 106 minutes and the half-life is 53 minutes.
$$ \text{Number of half-lives} = \frac{106}{53} = 2 $$
- Calculate remaining mass After each half-life, the mass is halved. So, after $n$ half-lives, the remaining mass is the initial mass divided by $2^n$. Initial mass is 54,544 grams.
$$ \text{Remaining mass} = \frac{\text{Initial mass}}{2^{\text{Number of half-lives}}} $$
- Plug in the values We have an initial mass of 54,544 grams, and 2 half lived have elapsed.
$$ \text{Remaining mass} = \frac{54544}{2^2} = \frac{54544}{4} $$
- Solve for the remaining mass
$$ \frac{54544}{4} = 13636 $$
$13636$
More Information
The half-life of a radioactive isotope is the time it takes for half of the atoms in a sample to decay.
Tips
A common mistake is to only divide by 2 once, forgetting that the mass halves with each half-life, not just once for the entire time period. Another mistake would be to divide the half-life by the total time.
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