To determine how many moles are contained in [tex]\(3.131 \times 10^{24}\)[/tex] particles, we use Avogadro's number, which is [tex]\(6.022 \times 10^{23}\)[/tex] particles per mole. Here is the step-by-step process to find the number of moles:
1. Understand the relationship: Avogadro's number tells us how many particles (atoms, molecules, etc.) are in one mole of a substance. This number is given as [tex]\(6.022 \times 10^{23}\)[/tex] particles per mole.
2. Set up the formula: The number of moles ([tex]\(n\)[/tex]) can be found using the formula:
[tex]\[
n = \frac{\text{Number of particles}}{\text{Avogadro's number}}
\][/tex]
3. Insert the given values:
[tex]\[
n = \frac{3.131 \times 10^{24} \text{ particles}}{6.022 \times 10^{23} \text{ particles/mole}}
\][/tex]
4. Perform the division: To find the exact number of moles, divide [tex]\(3.131 \times 10^{24}\)[/tex] by [tex]\(6.022 \times 10^{23}\)[/tex]:
[tex]\[
n = \frac{3.131}{6.022} \times 10^{24 - 23}
\][/tex]
5. Simplify the calculation: This simplifies to:
[tex]\[
n = \frac{3.131}{6.022} \times 10^1
\][/tex]
[tex]\[
n \approx \frac{3.131}{6.022} \times 10
\][/tex]
6. Calculate the approximate value: Carry out the division:
[tex]\[
n \approx 0.5199 \times 10
\][/tex]
[tex]\[
n \approx 5.199
\][/tex]
So, the calculated number of moles is approximately [tex]\(5.199\)[/tex].
7. Compare with the given options:
- A. [tex]\(5.199 \text{ mol}\)[/tex]
- B. [tex]\(18.85 \text{ mol}\)[/tex]
- C. [tex]\(0.5199 \times 10^{23} \text{ mol}\)[/tex]
- D. [tex]\(1.885 \times 10^{47} \text{ mol}\)[/tex]
The calculation shows that the number of moles is closest to the option A [tex]\(5.199 \text{ mol}\)[/tex].
Therefore, the correct answer is A.
