The half-life of a radioactive kind of protactinium is 27 days. How much will be left after 54 days, if you start with 488 grams of it?

Steps to Solve

1. Identify Half-Lives Over Time

First, we need to determine how many half-lives fit into the 54 days. The half-life of protactinium is 27 days.

We calculate the number of half-lives:

$$ \text{Number of half-lives} = \frac{54 \text{ days}}{27 \text{ days/half-life}} = 2 $$

2. Calculate Remaining Amount

Next, we apply the half-life formula to find the remaining amount of substance after 2 half-lives. The formula used is:

$$ \text{Remaining Amount} = \text{Initial Amount} \times \left(\frac{1}{2}\right)^{n} $$

Where ( n ) is the number of half-lives.

Plugging in the values:

$$ \text{Remaining Amount} = 488 \text{ grams} \times \left(\frac{1}{2}\right)^{2} $$

3. Perform the Calculation

Calculating the expression:

$$ \text{Remaining Amount} = 488 \text{ grams} \times \left(\frac{1}{2}\right)^{2} = 488 \text{ grams} \times \frac{1}{4} $$

4. Final Calculation of Remaining Amount

Calculating the final amount:

$$ \text{Remaining Amount} = 488 \text{ grams} \div 4 = 122 \text{ grams} $$