How much of a radioactive kind of promethium will be left after 20 days if you start with 9,056 grams and the half-life is 5 days?
Understand the Problem
The question is asking how much of a radioactive isotope of promethium will remain after 20 days, given an initial quantity and the half-life of the substance. We can approach this by using the formula for radioactive decay which takes the initial amount and reduces it by half for each half-life that passes.
Answer
$566$ grams
Answer for screen readers
The remaining amount of promethium after 20 days is $566$ grams.
Steps to Solve
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Determine the number of half-lives
First, find how many half-lives fit into the 20 days. Since the half-life is 5 days, we calculate: $$ \text{Number of half-lives} = \frac{20 \text{ days}}{5 \text{ days/half-life}} = 4 $$ -
Apply the half-life formula
The remaining amount after a certain number of half-lives can be calculated using the formula:
$$ \text{Remaining Amount} = \text{Initial Amount} \times \left(\frac{1}{2}\right)^{n} $$
where ( n ) is the number of half-lives. -
Substitute initial values into the formula
Now, substitute the initial amount (9,056 grams) and ( n ) (4 half-lives) into the formula:
$$ \text{Remaining Amount} = 9,056 \times \left(\frac{1}{2}\right)^{4} $$ -
Calculate the remaining amount
Evaluate the equation:
$$ \text{Remaining Amount} = 9,056 \times \left(\frac{1}{16}\right) = 9,056 \div 16 = 566 $$
The remaining amount of promethium after 20 days is $566$ grams.
More Information
After 20 days, 566 grams of the radioactive isotope remains. This process applies generally to any substance with a known half-life, illustrating how rapidly decay occurs over time.
Tips
- Miscalculating the number of half-lives: Ensure to divide the total time by the half-life accurately.
- Not using the correct formula: It's essential to use the correct radioactive decay formula.
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