To determine how much cesium remains from a 10 g sample after 6 years, given that cesium has a half-life of 2 years, we need to follow these steps:
1. Understand the Half-Life Concept:
The half-life is the amount of time it takes for half of a substance to decay. For cesium, this is 2 years.
2. Calculate the Number of Half-Lives:
We need to determine how many half-lives have passed in 6 years.
[tex]\[
\text{Number of half-lives} = \frac{\text{Time elapsed}}{\text{Half-life}}
\][/tex]
Given:
[tex]\[
\text{Time elapsed} = 6 \text{ years}
\][/tex]
[tex]\[
\text{Half-life} = 2 \text{ years}
\][/tex]
[tex]\[
\text{Number of half-lives} = \frac{6}{2} = 3
\][/tex]
So, 3 half-lives have passed in 6 years.
3. Determine the Remaining Amount of Cesium:
For each half-life, the amount of the substance reduces by half. Starting with an initial amount of 10 g, we need to halve this amount three times due to the three half-lives that have passed.
- After the first half-life:
[tex]\[
10 \text{ g} \times \frac{1}{2} = 5 \text{ g}
\][/tex]
- After the second half-life:
[tex]\[
5 \text{ g} \times \frac{1}{2} = 2.5 \text{ g}
\][/tex]
- After the third half-life:
[tex]\[
2.5 \text{ g} \times \frac{1}{2} = 1.25 \text{ g}
\][/tex]
So, after 6 years (which is equivalent to three half-lives), the remaining amount of cesium is [tex]\(1.25\)[/tex] g.
Therefore, the correct answer is:
D. [tex]\(1 \frac{1}{4} \text{ g}\)[/tex]
