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 is asking how much of a radioactive iodine sample will remain after a specific period, given its initial amount and half-life. We will need to calculate the number of half-lives that fit into the total time and use that to determine the remaining quantity.
Answer
$13,636$ grams
Answer for screen readers
The remaining amount of radioactive iodine after 106 minutes is 13,636 grams.
Steps to Solve
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Calculate the number of half-lives To find how many half-lives fit into the total time, divide the total time by the half-life.
The formula is:
$$ \text{Number of half-lives} = \frac{\text{Total time}}{\text{Half-life}} $$
Substituting the values:
$$ \text{Number of half-lives} = \frac{106 \text{ minutes}}{53 \text{ minutes}} = 2 $$ -
Determine the remaining mass The remaining amount of a radioactive substance after a number of half-lives can be calculated using the formula:
$$ \text{Remaining mass} = \text{Initial mass} \times \left(\frac{1}{2}\right)^{\text{Number of half-lives}} $$
Plugging in the values:
$$ \text{Remaining mass} = 54,544 \text{ grams} \times \left(\frac{1}{2}\right)^{2} $$ -
Calculate the final amount Now, compute the remaining mass:
$$ \text{Remaining mass} = 54,544 \text{ grams} \times \frac{1}{4} = 13,636 \text{ grams} $$
The remaining amount of radioactive iodine after 106 minutes is 13,636 grams.
More Information
The concept of half-life is important in understanding radioactive decay, as it shows how quickly a substance loses half of its mass over specified periods. In this case, we figured out that after two half-lives, the remaining mass was one-fourth of the original.
Tips
- Miscalculating the number of half-lives: Ensure to accurately divide the total time by the half-life to get the correct count of half-lives.
- Forgetting to square the fraction: It's essential to apply the exponent correctly to the fraction when calculating the remaining mass.
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