Radioactive Element Half-Life Explained

This question deals with the concept of radioactive decay, specifically how the amount of a radioactive substance changes over time based on its half-life period. The half-life is the time it takes for half of the radioactive atoms in a sample to decay.

Calculating Radioactive Decay Over Time

We are given:

• Initial amount of the radioactive element: 1 g
• The half-life period ($T_{1/2}$): 140 days
• The total time elapsed ($t$): 560 days

To find out how much of the element remains after 560 days, we first need to determine how many half-lives have occurred during this period.

Determining the Number of Half-Lives

The number of half-lives ($n$) can be calculated using the formula:

$$ n = \frac{t}{T_{1/2}} $$

Substituting the given values:

$$ n = \frac{560 \text{ days}}{140 \text{ days}} $$

$$ n = 4 $$

So, 4 half-lives have passed.

Calculating the Remaining Amount of the Element

After each half-life, the amount of the radioactive element reduces by half. The remaining amount can be calculated using the formula:

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

Using the initial amount of 1 g and the calculated number of half-lives ($n=4$):

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

First, calculate $\left(\frac{1}{2}\right)^4$:

$$ \left(\frac{1}{2}\right)^4 = \frac{1^4}{2^4} = \frac{1}{16} $$

Now, multiply this by the initial amount:

$$ \text{Remaining Amount} = 1 \text{ g} \times \frac{1}{16} $$

$$ \text{Remaining Amount} = \frac{1}{16} \text{ g} $$

Therefore, after 560 days, 1 g of the radioactive element will reduce to 1/16 g.